Calculator Techniques for Solving Progression Problems

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

Shift

SHIFT

Mode

MODE

Alpha

ALPHA

Stat

SHIFT 1[STAT]

AC

AC

Name Key Operation

(Sigma)

SHIFT log

Solve

SHIFT CALC

Logical equals

ALPHA CALC

Exponent

x[]

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.

HideClick here to show or hide the solution

common difference

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 Among the many STAT type, why A+BX?

The formula an = am + (n - m)d is linear in n. In calculator, we input n at X column and an at Y

column. Thus our X is linear representing the variable n in the formula.

Bring your calculator to Linear Regression in STAT mode:

MODE 3:STAT 2:A+BX and input the coordinates.

X (for n) Y (for an)

6 12

30 180

To find the first term:

AC 1 SHIFT 1[STAT] 7:Reg 5:y-caret and 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-caret and 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

Sum of the first 60 terms:

AC SHIFT log[] ALPHA )[X] SHIFT 1[STAT] 7:Reg 5:y-caret

SHIFT )[,] 1 SHIFT )[,] 60 )

The calculator will display (Xy-caret,1,60) then press [=].

(Xy-caret,1,60) = 11010 answer

Sum from 12th to 37th terms,

(Xy-caret,12,37) = 3679 answer

Another way to solve for the sum is to use the calculation outside the STAT mode. 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:COMP then 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

ShowClick here to show or hide the solution

Solution by Calculator

Why AB^X?

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.

MODE 3:STAT 6:AB^X

X Y

1 2

2 6

3 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,

AC SHIFT log[] ALPHA )[X] SHIFT 1[STAT] 7:Reg 5:y-caret

SHIFT )[,] 1 SHIFT )[,] 10 )

The calculator will display (Xy-caret,1,10) then press [=].

(Xy-caret,1,10) = 59048 answer

You may also sove the sum outside the STAT mode

(MODE 1:COMP then 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 Y

1 6

2 3

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.

- See more at: http://www.mathalino.com/blog/romel-verterra/solving-progression-problems-

calculator#sthash.HdtzRdds.dpuf