Challenging Problems in Sequences and Series- Special Hl Level Sums

SPECIAL HL GRADE SUMS SEQUENCES AND SERIES FROM: SRIRAMAN .IYER. HEAD, DEPT OF MATHEMATICS IB PROGRAMS, MUMBAI .INDIA EMAIL:[email protected]

(1) if 3, log y x, 3 log zy, 7 logxz are in Arithmetic Sequence Prove that (a) X 18 = y21 = z28 (b) Find x in terms of Y and Z separately

(2) If the ratio of Sum of first m terms and Sum of first n terms of an Arithmetic series is given by 𝑚 2 ( ) 𝑛

(a) Find the ratio of the mth term and nth term (b) Prove that the ratio never contains Even Numerator and Denominator for any value of m and n, where m and n are positive integers (c).if u = f(m) and v = f(n) , then find the Derivative of u and v with respect to its independent variables

(3) If the (m+1) th , (n+1) th and ( (r+1) th terms of an ARITHMETIC SEQUENCE are in GEOMETRIC SEQUENCE. a.Find the ratio between common difference and first term b. Prove that the above ratio is negative for any value of n if 2𝑚𝑟 𝑚 +𝑟

=n

(4) A sequence is given by 2+5+12+31+86+……….. (a) Find the nth term of the above sequence (b) Find the sum to n terms of the above sequence (c) Sn = f(n) , then find S’(n) (d) Find the sum till 20 terms

(5) (a) Find the ratio of Sum of n terms of sequences of “Sum of Natural numbers” and “Sum of Squared Natural Numbers” (b). Find the ratio of (a) for the first 10 terms (c) if f(n) represents the ratio (a), and g(n) represents the “ Sum of cubes of first n natural numbers” , then find the composite function of f and g (d) Check the commutative law of composite function of f and g

(6) If Y = x+ x3+x5+….. is a infinite Geometric sequence , find a.Sum of first 10 terms for x>1 b. Find the value of Y c. If Y = f(x) , find f’(x) d. Find the f-1(x)

(7) If f(x) = 𝑛𝑖=1 𝑖 log 𝑥 a.Find f(x). b.Find f(x) at x=10 c.Find f(x) at x=n=10 d.Find the inverse function of f(x) e.Find f’(1) at n=100

(8) Find the sum of n terms of 𝑥 (𝑥 2 ) + 1−(𝑥 2 ) 1−(𝑥 4 )

(a) (b) (c) (d)

+

(𝑥 4 ) 1−(𝑥 8 )

+ ……….

Find the nth term Find the sum of first n terms Find the sum when n = 4 If tn= f(n), find f’(1)

(9) In an Arithmetic Sequence, if m.tm=n.tn a.Prove that t(m+n)=0 b.Find the ratio of 5th term and 10th term

(10) A Series is given by 7+77+777+……. = S a.Find the nth term of the Series b.Find S for the first n terms c.Find S for first 20 terms using (a) d.If S = f(n), then find f’(n) d.find the n