A certain sequence is defined by the following rule: $S_n=k(S_{n-1})$ , where $k$ is a constant. If $S_1=2$ and $S_{13}=72$ , what is the value of $S_7$ ?

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Comments

AthinaAug 13

Took me a sec to realize this is just a geometric sequence in disguise. Once you see k^12 = 36, the rest falls out nicely.

NikhilAug 26

Yeah same, I was overcomplicating it with logs at first lol

BrianAug 18

Wait how did you get 36? I got 36 after dividing but then got stuck on what to do with it

NganMay 20

Is the trap answer E? Felt like they wanted people to panic-add. 655-705 level for sure.

SriMay 27

Haha I almost picked that until I realized you need k^6 not k^12

TaeikJul 25

Bit confused on why we take the 6th root instead of just halving the exponent. Can someone explain the exponent step?

ChristopherJul 30

S13 = S1 * k^12, and S7 = S1 * k^6. So S7 is basically the geometric mean between S1 and S13.

BijoyApr 15

Nice question, tests whether you actually understand the ratio pattern rather than just plugging numbers. 5/5 difficulty.

What this question tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.

A certain sequence is defined by the following rule: Sn=k(Sn−1)S_n=k(S_{n-1})Sn​=k(Sn−1​), where kkk is a constant. If S1=2S_1=2S1​=2 and S13=72S_{13}=72S13​=72, what is the value of S7S_7S7​?