Problem Solving

The infinitive sequence a1,a2.....,an... is such that a1=2, a2=-3, a3=5, a4=-1, and an=a(n-4) for n>4. What is the sum of the first 97 terms of the sequence?
72
74
75
78
80
Thanks

This is a periodic sequence. The sum of one period is 2+(-3)+5+(-1)=3. There are 24 periods till a_96, inclusive. Add 24 periods and a_97 for the total sum.

alex.gellatly wrote:The infinitive sequence a1,a2.....,an... is such that a1=2, a2=-3, a3=5, a4=-1, and an=a(n-4) for n>4. What is the sum of the first 97 terms of the sequence?
72
74
75
78
80
Thanks

a� = 2, a₂ = -3, a₃ = 5, a₄ = -1...

According to the formula, a(n) = a(n-4).
Thus:
a₅ = a� = 2
a₆ = a₂ = -3
a₇ = a₃ = 5
a₈ = a₄ = -1.
And so on.

The sequence repeats in a CYCLE OF 4: 2, -3, 5, -1...2, -3, 5, -1...
The sum of each cycle = 2 + (-3) + 5 + (-1) = 3.
Since 96/4 = 24, there will be total of 24 cycles between 1 and 97.
Sum of 24 cycles = 24*3 = 72.
a₉₇ will be the first term of a new cycle (2).
Adding a₉₇ = 2 to the sum of the 24 cycles, we get:
Sum of all 97 terms = 72+2 = 74.

The correct answer is B.