ISI2016-DCG-19

Question

The expression $3^{2n+1}+2^{n+2}$ is divisible by $7$ for

• A. all positive integer values of $n$
• B. all non-negative integer values of $n$
• C. only even integer values of $n$
• D. only odd integer values of $n$

Answer 1

So, option $B$ is the correct answer.

PS: In the last line of scanned paper, it is $n\geq 0$

Comments

nice :) but n=0 is also satisfied. right ?

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You checked by substituting $n$ directly?

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Yes correct.

Actually I was trying to learn to prove by induction while solving this problem and hence forget the simple objective check :D

Answer 2

Correct Answer - Option (B)

Let, $X = 3^{2n+1} + 2^{n+2} = 3(3)^{2n} + 4(2)^{2n}$

If we were to go from options, 

The edge cases to check here that would distinguish each option would be, 

$n \in \{0, 1,2\}$

now,

putting $n=0$, $3(1) + 4(1) = 7$  which is divisible by 7. So, that makes option (D) and (A) wrong.

putting $n=1$, $3(9) + 4(2) = 35$  which is divisible by 7. So, that makes option (C) wrong.

No need to check further since (B) is the only remaining option.

However, even for $n=2$, $3(81) + 4(4) = 243 + 16 = 259$ which is divisible by 7.

Here, option (B) is the only one that contains $\{0,1,2\}$, so it's the only correct choice.