The question is about sum of all terms. Four terms are given in which some of them are in AP and some of them are in GP, we need to find out the sum of all the terms. With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. CAT Exam tests the idea of progression often in the CAT Quantitative Aptitude section and this could also be tested in DI LR section of the CAT Exam as a part of a puzzle.

Question 20: Find the sum of all the terms, If the first 3 terms among 4 positive 2-digit integers are in A.P and the last 3 terms are in G.P. Moreover the difference between the first and last term is 40.

- 108
- 172
- 124
- 196

172

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Let the numbers be x-a, x, x+a

∴ For last 3 terms to be in G.P:-

(x+a)^{2} = x(x-a+40)

x^{2}+a^{2}+2ax = x^{2}-ax+40x

a^{2} = -3ax + 40x

=> x = \\frac{a^2}{(40-3a)}\\)

Now, since x is a positive integer, putting different integral value of a .

Only a = 12 gives a integer value of x = 36

(Note, a=b also give integral value x=b but then the first term becomes 0)

∴ x = 36, a = 12

∴ x – a = 24

x = 36

x+a = 48

x-a+40 = 64

∴ Sum of all the terms = 64+48+36+24 = 172

The terms are - 24,36,48,64

The question is **"Find the sum of all the terms, If the first 3 terms among 4 positive 2-digit integers are in A.P and the last 3 terms are in G.P."**

Choice B is the correct answer.

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