Calculator Techniques for Solving Progression Problems
January 8, 2017 | Author: Richard Regidor | Category: N/A
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Calculator Techniques for Solving Progression Problems
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This is the first round for series of posts about optimizing the use of calculator in solving math problems. The calculator techniques I am presenting here has been known to many students who are about to take the engineering board exam. Using it will save you plenty of time and use that time in analyzing more complex problems. The following models of CASIO calculator may work with these methods: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus. This post will focus on progression progression. To illustrate the use of calculator, we will have sample problems to solve. But before that, note the following calculator keys and the corresponding operation: Name
Key
Operation
Name
Shift
SHIFT
Σ (Sigma)
SHIFT → log
Mode
MODE
Solve
SHIFT → CALC
Alpha
ALPHA
Logical equals
ALPHA → CALC
Stat
SHIFT → 1[STAT]
Exponent
x[]
AC
AC
Problem: Arithmetic Progression The 6th term of an arithmetic progression is 12 and the 30th term is 180. 1. What is the common difference of the sequence? 2. Determine the first term? 3. Find the 52nd term. 4. If the nth term is 250, find n. 5. Calculate the sum of the first 60 terms. 6. Compute for the sum between 12th and 37th terms, inclusive. Traditional Solution For a little background about Arithmetic Progression, the traditional way of solving this problem is presented here. Click here to show or hide the solution
→ common difference
Key
Operation
→ first term
→ 52nd term
→ 40th term, a40 = 250 Sum of AP is given by the formula
Sum of the first 60 terms → answer Sum between 12th and 37th terms, inclusive.
→ answer
Calculator Technique for Arithmetic Progression Bring your calculator to Linear Regression in STAT mode: MODE → 3:STAT → 2:A+BXand input the coordinates.
Among the many STATtype, why A+BX? The formula an = am + (n - m)d is linear in n. In
X (for n)
Y (for an)
calculator, we input n at X column and an at Y column.
6 30
12 180
Thus our X is linear representing the variable n in the formula.
To find the first term: AC → 1 SHIFT → 1[STAT] → 7:Reg → 5:y-caretand calculate 1y-caret, be sure to place 1 in front of y-caret. 1y-caret= -23
→ answer for the first term
To find the 52nd term, and again AC → 52 SHIFT → 1[STAT] → 7:Reg → 5:y-caretand make sure you place 52 in front of y-caret. 52y-caret= 334
→ answer for the 52nd term
To find n for an = 250, AC → 250 SHIFT → 1[STAT] → 7:Reg → 4:x-caret 250x-caret= 40
→ answer for n
To find the common difference, solve for any term adjacent to a given term, say 7th term because the 6th term is given then do 7y-caret- 12 = 7 for d. For some fun, randomly subtract any two adjacent terms like 18y-caret- 17y-caret, etc. Try it! Sum of Arithmetic Progression by Calculator Bring the your calculator to Quadratic Regression in STAT mode MODE → 3:STAT → 3:_+cX2
Note that for the given AP, a1 = -23, a2 = -16, and a3 = -9. Input three coordinates X 1 2 3
Y -23 -23-16 -23-16-9
Sum of the first 60 terms: (AC → 60 SHIFT → 1[STAT] → 7:Reg → 6:y-caret) 60y-caret= 11010
Why MODE → 3:STAT → 3:_+cX2? The formula S = ½n[ 2a1 + (n - 1)d ] for sum of arithmetic progression is quadratic in n. In our calculator, we input n in the X column and the sum at the Y column.
Sum from 12th to 37th terms, use SHIFT → 1[STAT] → 7:Reg → 6:y-carettwice 37y-caret - 11y-caret= 3679 Another way to solve for the sum is to use the Σ calculation. The concept is to add each term in the progression. Any term in the progression is given by an = a1 + (n - 1)d. In this problem, a1 = -23 and d = 7, thus, our equation for an is an = -23 + (n - 1)(7). Reset your calculator into general calculation mode: MODE → 1:COMPthen SHIFT → log. Sum of first 60 terms: (-23 + (ALPHA X - 1) × 7)= 11010
Or you can do (-23 + 7 ALPHA X)= 11010 which yield the same result.
Sum from 12th to 37th terms (-23 + (ALPHA X - 1) × 7)= 3679
Or you may do (-23 + 7 ALPHA X)= 3679
Calculator Technique for Geometric Progression Problem Given the sequence 2, 6, 18, 54, ... 1. Find the 12th term 2. Find n if an = 9,565,938. 3. Find the sum of the first ten terms. Traditional Solution Click here to show or hide the solution
Solution by Calculator MODE → 3:STAT → 6:A·B^X
Why A·B^X?
X 1 2 3
The nth term formula an = a1rn – 1 for geometric progression is exponential in form, the variable n in the formula is the X equivalent in the calculator.
Y 2 6 18
To solve for the 12th term AC → 12 SHIFT → 1[STAT] → 7:Reg → 5:y-caret 12y-caret= 354294
answer
To solve for n, AC → 9565938 SHIFT → 1[STAT] → 7:Reg → 4:x-caret 9565938x-caret= 15
answer
Sum of the first ten terms (MODE → 1:COMPthen SHIFT → log) Each term which is given by an = a1rn – 1. (2(3ALPHA X - 1))= 59048
answer
Or you may do (2 × 3ALPHA X)= 59048
Calculator Technique for Harmonic Progression
Problem Find the 30th term of the sequence 6, 3, 2, ... Solution by Calculator MODE → 3:STAT → 8:1/X
X 1 2 3
Y 6 3 2
AC → 30 SHIFT → 1[STAT] → 7:Reg → 5:y-caret 30y-caret= 0.2
answer
I hope you find this post helpful. With some practice, you will get familiar with your calculator and the methods we present here. I encourage you to do some practice, once you grasp it, you can easily solve basic problems in progression. If you have another way of using your calculator for solving progression problems, please share it to us. We will be happy to have variety of ways posted here. You can use the comment form below to do it. Tags: scientific calculatorcalculator techniqueCASIO calculatorarithmetic progression by calculatorgeometric progression by calculatorharmonic progression by calculator
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