The sequence a1, a2, ..., an, ... is such that a1 = 3, a2 = 5, and an

Question

The sequence a_1 , a_2 , ..., a_n , ... is such that a_1 = 3 , a_2 = 5 , and a_n = a_{n-1} - a_{n-2} for all integers n ≥ 3 . What is the sum of the first 100 terms of the sequence? (A) 2 (B) 3 (C) 7 (D) 8 (E) 10 2024-01-28_15-48-33.png

gmatophobia · Accepted Answer

Let's write few terms of the sequence to establish a pattern

a_1 = 3

a_2 = 5

a_3 = 5 - 3 = 2

a_4 = 2 - 5 = -3

a_5 = -3 - 2 = -5

a_c = -5 + 3 = -2

a_6 = -2 + 5 = 3

a_7 = 3 + 2 = 5

.
.

So, we can see the pattern repeating after every  6^{	ext{th}}  term.

Sum of the first 6 terms =  3 + 5 + 2 - 3 - 5 -2 = 0

100 = 6*16 + 4

Hence, the sum of the first 96 terms = 0

a_{97} = 3

a_{98} = 5

a_{99} = 2

a_{100} = -3

Sum of the first 100 terms =  3 + 5 + 2 - 3 = 7

Option C

filippocarta01 · Answer

I don't get why the sixth term in the pattern is Ac and not A6. 
Thank you very much

Kinshook · Answer

filippocarta01

Given: The sequence  a_1, a_2, …, a_n, ….  is such that  a_1 = 3, a_2 = 5 , and   a_n = a_{n-1} – a_{n-2}  for all integers n ≥3. 
Asked: What is the sum of the first  100  terms of the sequence?

a_1 = 3, a_2 = 5, a_3 = 5-3=2, a_4=2-5=-3 ,  a_5=-3-2=-5,a_6=-5-(-3)=-2;a_7=-2-(-5)=3,a_8=3-(-2)=5

Sum of every 6 terms is 0;
100 = 6*16 + 4
Sum of 96 terms = 0
Sum of last 4 terms = 3+5+2-3=7

Sum of first 100 terms = 0+7 = 7

IMO C

Bhavesh1291997 · Answer

for all integers n≥3; doesn't this mean we have to find sum of all 100 terms that start from A3 i.e A3 to A103
Answer would be diff in that case

Bhavesh1291997 · Answer

for all integers n≥3; doesn't this mean we have to find sum of all 100 terms that start from A3 i.e A3 to A103
Answer would be diff in that case

Bunuel · Answer

No. “For all integers n ≥ 3” only specifies for which values of n the formula applies. It does not redefine which terms belong to the sequence. The sequence is explicitly defined starting with a1 and a2, and the rule is used to generate a3, a4, and so on. Therefore, “the first 100 terms” means a1 through a100, not a3 through a103.

The sequence a1, a2, ..., an, ... is such that a1 = 3, a2 = 5, and an

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The sequence a1, a2, ..., an, ... is such that a1 = 3, a2 = 5, and an

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