The infinite sequence a_1, a_2, , a_n, is such that...

Question

The infinite sequence a_1, a_2, , a_n, is such that a_1=2, a_2=-3, a_3=5, a_4=-1, and a_n=a_n- _4 for n>4. What is the sum of the first 97 terms of the sequence?

GMAT Panda · Accepted Answer

74 We note that the sequence repeats every 4 terms due to the definition of a_n. a_n=a_n-4 We find how many full 4-term cycles are contained in the first 97 terms and the number of extra terms. 97 4 = 24 remainder 1 We calculate the sum of one full 4-term cycle. 2 + (-3) + 5 + (-1) = 3 We multiply the number of full cycles by the cycle sum to get the total from all full cycles. 24 3 = (20 + 4) 3 = (20 3) + (4 3) = 60 + 12 = 72 We add the contribution from the single extra term (the first term of the next cycle). 72 + 2 = 74

Explanation	Calculations	Help
$We note that the sequence repeats every 4 terms due to the definition of a_{n} .$	$a_{n} = a_{n - 4}$
$We find how many full 4-term cycles are contained in the first 97 terms and the number of extra terms.$	$97 \div 4 = 24 remainder 1$
$We calculate the sum of one full 4-term cycle.$	$2 + (- 3) + 5 + (- 1) = 3$
$We multiply the number of full cycles by the cycle sum to get the total from all full cycles.$	$24 \times 3 = (20 + 4) \times 3 = (20 \times 3) + (4 \times 3) = 60 + 12 = 72$
$We add the contribution from the single extra term (the first term of the next cycle).$	$72 + 2 = 74$

Question

Options:

Steps

Final Answer

Question

Options:

Steps

Final Answer

Related questions