progression & series

Brain Teasures

Progression and Series

By Abhijit kumar Jha

EXERCISE–I Q.1

If the 10th term of an HP is 21 & 21st term of the same HP is 10, then find the 210th term.

Q.2

Show that ln (4 × 12 × 36 × 108 × .............. up to n terms) = 2n ln 2 +

Q.3

There are n AM’s between 1 & 31 such that 7th mean : (n  1)th mean = 5 : 9, then find the value of n.

Q.4

Find the sum of the series , 7 + 77 + 777 + ..... to n terms.

Q.5

Express the recurring decimal 0.1576 as a rational number using concept of infinite geometric series.

Q.6

Find the sum of the n terms of the sequence

Q.7

The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369. The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

Q.8

If the pth, qth & rth terms of an AP are in GP . Show that the common ratio of the GP is

Q.9

If one AM ‘a’ & two GM’s p & q be inserted between any two given numbers then show that p3+ q3 = 2 apq .

Q.10

1

1 1  1 2

4



2

1 2  2 2

4



3

n ( n  1) ln 3 2

1  3  34 2

 ........

qr pq

.

The sum of n terms of two arithmetic series are in the ratio of (7 n + 1) : (4 n + 27) . Find the ratio of their nth term. If S be the sum , P the product & R the sum of the reciprocals of a GP , find the value of P 2  R  . n

Q.11 Q.12

Q.13

 S

The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is formed by multiplying each of the first 2n terms by the next term. Show that the sum of the new series is ( 4n 2  1)(a 2  b 2 )  ( 4n 2  2)ab . 6n In an AP of which ‘a’ is the Ist term, if the sum of the Ist p terms is equal to zero , show that the sum of the next q terms is  a (p + q) q/(p  1).

Q.14(a) The interior angles of a polygon are in AP. The smallest angle is 120° & the common difference is 5°. Find the number of sides of the polygon. (b) The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygon if its largest interior angle is 172°. Q.15

An AP & an HP have the same first term, the same last term & the same number of terms ; prove that the product of the rth term from the beginning in one series & the rth term from the end in the other is independent of r.

Q.16

Find three numbers a , b , c between 2 & 18 such that ; (i) their sum is 25 (ii) the numbers 2, a, b are consecutive terms of an AP & (iii) the numbers b , c , 18 are consecutive terms of a GP .

Q.17

Given that ax = by = cz = du & a , b , c , d are in GP, show that x , y , z , u are in HP .

Q.18

In a set of four numbers, the first three are in GP & the last three are in AP , with common difference 6. If the first number is the same as the fourth , find the four numbers.

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Brain Teasures

Progression and Series

By Abhijit kumar Jha

Find the sum of the first n terms of the sequence : 1  2  1  1   3  1  1   4  1  1   .........       2

Q.19

n

n

3

n

Q.20

Find the nth term and the sum to n terms of the sequence : (i) 1 + 5 + 13 + 29 + 61 + ...... (ii) 6 + 13 + 22 + 33 + .......

Q.21

The AM of two numbers exceeds their GM by 15 & HM by 27 . Find the numbers.

Q.22

The harmonic mean of two numbers is 4. The airthmetic mean A & the geometric mean G satisfy the relation 2 A + G² = 27. Find the two numbers.

Q.23

Sum the following series to n terms and to infinity : 1 1 1    ......... 1 . 4 .7 4 .7 .10 7 .10 .13

(ii)

(iii)



(iv)

1

n

Q.24

r 1

4r  1 2

r 1

1 1 .3 1 .3 . 5    ........... . 4 4 .6 4 .6 .8

Find the value of the sum

  rs 2r 3s where rs is zero if r  s & rs is one if r = s. n

(a) (b) Q.25

 r (r + 1) (r + 2) (r + 3) n

(i)

n

r 1

s1

   1. n

i

j

i 1

j1

k 1

For or 0 <  < /2, if : x=





n0

n0



 cos2n  , y =  sin2n  , z =  cos2n  sin2n  then : Prove that

(i) xyz = xy + z

n0

(ii) xyz = x + y + z

EXERCISE–II Q.1 Q.2

The series of natural numbers is divided into groups (1), (2, 3, 4), (5, 6, 7, 8, 9), ...... & so on. Show that the sum of the numbers in the nth group is (n  1)3 + n3 . The sum of the squares of three distinct real numbers , which are in GP is S² . If their sum is  S, show that ²  (1/3 , 1)  (1 , 3) .

Q.3

If there be m AP’s beginning with unity whose common difference is 1 , 2 , 3 .... m . Show that the sum of their nth terms is (m/2) (mn  m + n + 1).

Q.4

If Sn represents the sum to n terms of a GP whose first term & common ratio are a & r respectively, then prove that S1 + S3 + S5 + ..... + S2n-1 =

an a r (1  r 2 n )  . 1  r (1  r) 2 (1  r)

Q.5

A geometrical & harmonic progression have the same pth, qth & rth terms a, b, c respectively. Show that a (b  c) log a + b (c  a) log b + c (a  b) log c = 0.

Q.6

A computer solved several problems in succession. The time it took the computer to solve each successive problem was the same number of times smaller than the time it took to solve the preceding problem. How many problems were suggested to the computer if it spent 63.5 min to solve all the problems except for the first, 127 min to solve all the problems except for the last one, and 31.5 min to solve all the problems except for the first two?

Q.7

If the sum of m terms of an AP is equal to the sum of either the next n terms or the next p terms of the same AP prove that (m + n) [(1/m)  (1/p)] = (m + p) [(1/m)  (1/n)] (n  p) http://akj259.worpress.com

Brain Teasures

Progression and Series

By Abhijit kumar Jha

If the roots of 10x3  cx2  54x  27 = 0 are in harmonic progression, then find c & all the roots.

Q.8

Q.9(a) Let a1, a2, a3 ...... an be an AP . Prove that : 1 1 1 1    ..........  a1 a n a 2 a n 1 a 3 a n2 a n a1

=

1 2 1 1 1   ..........     a1  a n  a1 a 2 a3 an 

(b) Show that in any arithmetic progression a1, a2, a3 ....... a1²  a2² + a3²  a4² + ...... + a²2K  1  a²2K = [K/(2 K  1)] (a1²  a²2K) . Q.10

Let Let

a1 , a2 , ........... , an , an+1 , ....... be an A.P. S1 = a1 + a2 + a3 + ............. + an S2 = an+1 + an+2 + ...............+ a2n S3 = a2n+1 + a2n+2 + .............+ a3n ................................................ ................................................ Prove that the sequence S1 , S2 , S3 , ........ is an arithmetic progression whose common difference is n2 times the common difference of the given progression.

Q.11

If a, b, c are in HP, b, c, d are in GP & c, d, e are in AP, Show that e = ab²/(2a  b)² .

Q.12 (i) (ii) (iii)

If a, b, c, d, e be 5 numbers such that a, b, c are in AP ; b, c, d are in GP & c, d, e are in HP then: Prove that a, c, e are in GP . Prove that e = (2 b  a)²/a . If a = 2 & e = 18 , find all possible values of b , c , d .

Q.13

The sequence a1, a2, a3, ....... a98 satisfies the relation an+1 = an + 1 for n = 1, 2, 3, .........97 and has

 a 2k . 49

the sum equal to 4949. Evaluate

k 1

Q.14

If n is a root of the equation x² (1  ac)  x (a² + c²)  (1 + ac) = 0 & if n HM’s are inserted between a & c, show that the difference between the first & the last mean is equal to ac(a  c) .

Q.15

(a) (b)

Q.16

The value of x + y + z is 15 if a , x , y , z , b are in AP while the value of ; (1/x)+(1/y)+(1/z) is 5/3 if a , x , y , z , b are in HP . Find a & b . The values of xyz is 15/2 or 18/5 according as the series a , x , y , z , b is an AP or HP . Find the values of a & b assuming them to be positive integer .

An AP , a GP & a HP have ‘a’ & ‘b’ for their first two terms . Show that their (n + 2)th terms will be 2 n2 a 2 n2 n1 . in GP if b  n ba b 2 n a 2 n





1.3 3.5 5.7 7.9    ..........23 . 2 2 2 23 2 4

Q.17

Prove that the sum of the infinite series

Q.18

If there are n quantities in GP with common ratio r & Sm denotes the sum of the first m terms, show that the sum of the products of these m terms taken two & two together is [r/(r + 1)] [Sm] [Sm  1] .

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Brain Teasures Q.19

Progression and Series

By Abhijit kumar Jha

Find the condition that the roots of the equation x3 – px2 + qx – r = 0 may be in A.P. and hence solve the equation x3 – 12x2 + 39x –28 = 0.

Q.20

If ax2 + 2bx + c = 0 & a1x2 + 2 b1x + c1 = 0 have a common root & a/a1 , b/b1 , c/c1 are in AP, show that a1 , b1 & c1 are in GP.

Q.21

If a , b , c be in GP & logc a, logb c, loga b be in AP , then show that the common difference of the AP must be 3/2.

Q.22

If a1 = 1 & for n > 1 , an = an-1 +

Q.23

Sum to n terms :

1 a n 1

, then show that 12 < a75 < 15.

(i)

1 2x 3x2    ....... x  1 (x  1) (x  2) (x  1) (x  2) (x  3)

(ii)

a1 a2 a3    ....... 1  a 1 1  a 1  1  a 2  1  a 1  1  a 2  1  a 3 

Q.24

In a GP the ratio of the sum of the first eleven terms to the sum of the last eleven terms is 1/8 and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2. Find the number of terms in the GP.

Q.25

Given a three digit number whose digits are three successive terms of a G.P. If we subtract 792 from it, we get a number written by the same digits in the reverse order . Now if we subtract four from the hundred's digit of the initial number and leave the other digits unchanged, we get a number whose digits are successive terms of an A.P. Find the number.

EXERCISE–III Q.1 Q.2

Q.3 Q.4

For any odd integer n  1, n3  (n  1)3 + ...... + ( 1)n  1 l3 = ______ .

[ JEE ’96, 1]

x = 1+ 3a + 6a² + 10a3 + ..... a < 1 y = 1+ 4b + 10b² + 20b3 + ..... b < 1, find S = 1+ 3ab + 5(ab)² + .... in terms of x & y. [ REE ’96, 6 ] The real numbers x1, x2, x3 satisfying the equation x3  x² +  x +  = 0 are in A.P . Find the intervals in which  and  lie . [JEE ’96, 3] Let p & q be roots of the equation x2  2x + A = 0, and let r & s be the roots of the equation x2  18x + B = 0 . If p < q < r < s are in arithmatic progression, then A = ______, and B = ______. [ JEE ’97, 2 ]

Q.5

a, b, c are the first three terms of a geometric series. If the harmonic mean of a & b is 12 and that of b & c is 36, find the first five terms of the series. [ REE '98, 6 ]

Q.6 (a)

Select the correct alternative(s). [ JEE '98, 2 + 2 + 8 ] th Let Tr be the r term of an AP, for r = 1, 2, 3, .... If for some positive integers m, n we have Tm = (A)

(b)

1 1 & Tn = , then Tmn equals : n m

1 mn

1 1  m n

If x = 1, y > 1, z > 1 are in GP, then (A) AP

(c)

(B)

(B) HP

(C) 1

(D) 0

1 1 1 , , are in : 1  n x 1  n y 1  n z

(C) GP

(D) none of the above

Prove that a triangle ABC is equilateral if & only if tan A + tan B + tan C = 3 3 . http://akj259.worpress.com

Brain Teasures

Progression and Series





By Abhijit kumar Jha





Q.7(a) The harmonic mean of the roots of the equation 5  2 x2  4  5 x + 8 + 2 5 = 0 is (A) 2

(B) 4

(C) 6

(D) 8

(b) Let a1, a2,...., a10, be in A.P. & h1, h2, ....., h10 be in H.P. If a1 = h1 = 2 & a10 = h10 = 3 then a4 h7 is: (A) 2 (B) 3 (C) 5 (D) 6 [ JEE '99, 2 + 2 out of 200 ] Q.8

The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. If the inverse of its common ratio is an integer, find all possible values of the common ratio, n and the first terms of the series. [ REE '99, 6 ]

Q.9(a) Consider an infinite geometric series with first term 'a' and common ratio r . If the sum is 4 and the second term is 3/4, then : (A) a =

7 3 , r= 4 7

(B) a = 2 , r =

3 8

(C) a =

3 1 , r= 2 2

(D) a = 3 , r =

1 4

(b) If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation : (A) 0  M  1 (B) 1  M  2 (C) 2  M  3 (D) 3  M  4 [ JEE 2000, Screening, 1 + 1 out of 35 ] (c) The fourth power of the common difference of an arithmetic progression with integer entries added to the product of any four consecutive terms of it . Prove that the resulting sum is the square of an integer. [ JEE 2000, Mains, 4 out of 100 ] Q.10

Given that ,  are roots of the equation, A x2  4 x + 1 = 0 and ,  the roots of the equation, B x2  6 x + 1 = 0, find values of A and B, such that , ,  &  are in H.P. [ REE 2000, 5 out of 100 ]

Q.11

The sum of roots of the equation ax2 + bx + c = 0 is equal to the sum of squares of their reciprocals. Find whether bc2, ca2 and ab2 in A.P., G.P. or H.P.? [ REE 2001, 3 out of 100 ]

Q.12

Solve the following equations for x and y log2x + log4x + log16x + .................... = y 5  9  13............ ( 4y  1) = 4log4x 1  3  5.............. (2y  1)

[ REE 2001, 5 out of 100 ]

Q.13(a) Let  be the roots of x2 – x + p = 0 and  be the roots of x2 – 4x + q = 0. If are in G.P., then the integral values of p and q respectively, are (A) –2, –32 (B) –2, 3 (C) –6, 3 (D) –6, –32 (b) If the sum of the first 2n terms of the A.P. 2, 5, 8, ........... is equal to the sum of the first n terms of the A.P. 57, 59, 61, ........, then n equals (A) 10 (B) 12 (C) 11 (D) 13 (c) Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are (A) NOT in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D) H.P. [ JEE 2001, Screening, 1 + 1 + 1 out of 35 ] http://akj259.worpress.com

Brain Teasures

Progression and Series

By Abhijit kumar Jha

(d)

Let a1, a2 .......... be positive real numbers in G.P. For each n, let An, Gn, Hn, be respectively, the arithmetic mean, geometric mean and harmonic mean of a1, a2, a3, ...........an. Find an expression for the G.M. of G1, G2, .........Gn in terms of A1, A2 .............An, H1, H2, .........Hn. [ JEE 2001 3 (Mains); 5] 2 2 2 Q.14(a) Suppose a, b, c are in A.P. and a , b , c are in G.P. If a < b < c and a + b + c = , then the value of 2 a is (A)

1

(B)

2 2

1 2 3

(C)

1 1  2 3

1 1  2 2 [JEE 2002 (Screening), 3]

(D)

(b) Let a, b be positive real numbers. If a , A1 , A2 , b are in A.P. ; a , a1 , a2 , b are in G.P. and a , H1 , H2 , b are in H.P. , show that G1 G 2 A  A2 (2 a  b) ( a  2 b )  1  H1 H 2 H1  H 2 9ab

[ JEE 2002 , Mains , 5 out of 60 ]

Q.15

If a, b, c are in A.P., a2 , b2 , c2 are in H.P. , then prove that either a = b = c or a, b, 

Q.16

The first term of an infinite geometric progression is x and its sum is 5. Then

c form a G.P.. 2 [JEE-03, Mains-4 out of 60]

(A) 0 x  10 Q.17

(B) 0 < x < 10

(C) –10 < x < 0

(D) x > 10

[JEE 2004 (Screening)] If a, b, c are positive real numbers, then prove that [(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4. [JEE 2004, 4 out of 60]

Q.18(a) In the quadratic equation ax2 + bx + c = 0, if  = b2 – 4ac and  + , 2 + 2, 3 + 3 are in G.P. where ,  are the roots of ax2 + bx + c = 0, then (A)   0 (B) b = 0 (C) c = 0 (D)  = 0 [JEE 2005 (Screening)]  n  1  n+1  (2 – n – 2) where n > 1, and the runs scored in (b) If total number of runs scored in n matches is   4  the kth match are given by k·2n+1– k, where 1  k  n. Find n. [JEE 2005 (Mains), 2]

3 3 3 3 If A n           .......   1n 1   and Bn = 1 – A An, then find the minimum natural 4 4 4 4 number n0 such that Bn > An.  n > n0. [JEE 2006, 6] 2

Q.19

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3

n

Brain Teasures

Progression and Series

By Abhijit kumar Jha

ANSWER KEY EXERCISE–I Q 1. 1 Q 3. µ = 14 n+1 Q 4. S = (7/81){10  9n  10} Q 5. 35/222 Q 6. n (n + 1)/2 (n² + n + 1) Q 7. 27 Q 10. (14 n  6)/(8 n + 23) Q 11. 1 Q 14. (a) 9 ; (b) 12 Q 16. a = 5 , b = 8 , c = 12 Q 18. (8 ,  4 , 2 , 8) Q 19. n² n+1 n+2 Q 20. (i) 2  3 ; 2  4  3n (ii) n² + 4n + 1 ; (1/6) n (n + 1) (2n + 13) + n Q 21. 120 , 30 Q 22. 6 , 3 Q 23. (i) sn = (1/24)  [1/{6(3n + 1) (3n + 4) }] ; s = 1/24 (ii) (1/5) n (n + 1) (n + 2) (n + 3) (n + 4)  1 1.3.5.....(2n  1)(2n  1)  (iv) Sn = 2   ; S = 1 2.4.6......(2n )(2n  2)  2 Q 24. (a) (6/5) (6n  1) (b) [n (n + 1) (n + 2)]/6 (iii) n/(2n + 1)

EXERCISE–II Q 6. 8 problems , 127.5 minutes Q.8 C = 9 ; (3, 3/2 , 3/5) Q 12. (iii) b = 4 , c = 6 , d = 9 OR b =  2 , c =  6 , d =  18 Q.13 Q 15. (a) a = 1 , b = 9 OR b = 1 , a = 9 ; (b) a = 1 ; b = 3 or vice versa Q.19 2p3 – 9pq + 27r = 0; roots are 1, 4, 7 Q 23. (a) 1 

xn (x  1) (x  2) ..... (x  n)

Q 24. n = 38

(b) 1 

2499

1 (1  a 1 ) (1  a 2 ) ..... (1  a n )

Q 25. 931

EXERCISE–III 1 4

Q 1.

(2n  1) (n + 1)²

Q 2. S =

1  ab

(1  ab) 2

Where a = 1  x–1/3 & b = 1  y–1/4

Q3.

  (1/3) ;   (1/27)

Q 4.  3, 77 Q 6. (a) C (b) B

Q 5. 8, 24, 72, 216, 648 Q 7. (a) B (b) D

Q 9. (a) D

Q 10. A = 3 ; B = 8

Q 8. r = ± 1/9 ; n = 2 ; a = 144/180 OR r = ± 1/3 ; n = 4 ; a = 108 OR r = 1/81 ; n = 1 ; a = 160

Q 11. A.P.

(b) A



Q 12. x = 2 2 and y = 3



Q 13. (a) A, (b) C, (c) D , (d) A1, A2 ,............ A n   H1, H2 ,............ H n  Q14.

(a) D

Q.16

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B

Q.18

(a) C, (b) n = 7

1 2n

Q.19

n0 = 5