Track / Stage 3 / 22 of 260 #22 of 2000

Problem 22

AMC 10/12, early questions

Algebra Difficulty 3.0 Find the answer

Given a sequence $\{a_n\}$ , the sum of the first $n$ terms $S_n$ satisfies $S_n=2a_n-2$ .
(1) Find $a_1$ , $a_2$ , $a_3$ and from this conjecture the general formula for $a_n$ ;
(2) Use mathematical induction to prove the general formula for $\{a_n\}$ .

Your answer

Official solution

Solution:
(1) Since $S_n=2a_n-2$ ,
When $n=1$ , $a_1=2a_1-2$ , solving this gives $a_1=2$ .
When $n=2$ , $a_1+a_2=2a_2-2$ , solving this gives $a_2=4$ .
When $n=3$ , $a_1+a_2+a_3=2a_3-2$ , solving this gives $a_3=8$ .
Conjecture: $a_n=2^n$ .
(2) When $n=1$ , the conjecture obviously holds.
Assume when $n=k$ , the conjecture holds, i.e., $a_k=2^k$ .
Then when $n=k+1$ , $S_{k+1}=2a_{k+1}-2$ .
Therefore, $S_k+a_{k+1}=2a_{k+1}-2$ ,
Therefore, $2a_k-2+a_{k+1}=2a_{k+1}-2$ ,
Therefore, $a_{k+1}=2a_k=2\cdot2^k=2^{k+1}$ .
Therefore, when $n=k+1$ , the conjecture holds.
Therefore, $a_n=2^n$ .

Thus, the final answer is $\boxed{a_n=2^n}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.