GMAT Practice Question: The infinite sequence a_{1}, a_{2}, \ldots, a_{n}, \ldots is such that...
Question
The infinite sequence a_{1}, a_{2}, \ldots, a_{n}, \ldots 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?
- 72
- 74
- 75
- 78
- 80
Topics: patterns, sequences & series
Solution
Step 1
We note that the sequence repeats every 4 terms due to the definition of a_{n}.
a_{n}=a_{n-4}
Step 2
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 \text{ remainder } 1
Step 3
We calculate the sum of one full 4-term cycle.
2 + (-3) + 5 + (-1) = 3
Step 4
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
Step 5
We add the contribution from the single extra term (the first term of the next cycle).
72 + 2 = 74
Answer
74
