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Integer GP, 5 Terms All Under 100: Count Them | JEE

JEE Maths question with a full step-by-step solution.

Question

Let

a_1, a_2, a_3, a_4, a_5

be five terms of a geometric sequence satisfying

0<a_1<a_2<a_3<a_4<a_5<100

where each term is an integer. Then the number of such geometric progressions is

6

7

8

correct

10

Solution

Step 1: Let the ratio be

\dfrac{n}{m}

in lowest terms with

n>m

. For every term to be an integer,

a_1

must be a multiple of

m^4

; write

a_1=k\,m^4

. The terms are

k m^4,\ k m^3 n,\ k m^2 n^2,\ k m n^3,\ k n^4.

Step 2: The largest satisfies

kn^4<100

. If

n\ge 4

, then

n^4\ge 256>100

— impossible, so

n\le 3.

Step 3: The cases:

n=3,\ m=2:\ 81k<100\Rightarrow k=1:\ 16,24,36,54,81.

n=3,\ m=1:\ 81k<100\Rightarrow k=1:\ 1,3,9,27,81.

n=2,\ m=1:\ 16k<100\Rightarrow k=1,\ldots,6:\ \text{six progressions}.

Step 4: Total

=1+1+6=8.

Correct answer: (3)

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