Question

The recurrence relation an = –an – 1 + n – 1 with the initial condition a0 = 7

an = n − 1 − an − 1
= n − 1 − ((n − 1 − 1) − an − 2
= (n − 1) − (n − 2) + an − 2
= (n − 1) − (n − 2) + ((n − 2 − 1) − an − 3)
= (n − 1) − (n − 2) + (n − 3) − an − 3
= ⋅ ⋅ ⋅ continuing in the same manner
= (n − 1) − (n − 2) + · · · + (−1)n − 1(n − n) + (−1)nan − n
= _____
a.) 2n+ (−1)ⁿ /4 + (−1)ⁿ⋅ 7
b.) 2n− 1+ (−1)ⁿ /4 + (−1)ⁿ⋅ 7
c.) 2n− 1−(−1)ⁿ /4 + (−1)ⁿ
d.) 2n− 1+ (−1)ⁿ /4 − (−1)ⁿ

Answer 1

The general formula for an in the given recurrence relation is 2n - 1 + (-1)^n / 4 - (-1)^n.

Step-by-step explanation:

The given recurrence relation is an = –an – 1 + n – 1 with the initial condition a0 = 7. To find a general formula for an, we can simplify the expression:

an = (n - 1) - an - 1 = (n - 1) - ((n - 1) - 1) - an - 2 = (n - 1) - (n - 2) + an -2

Continuing in the same manner, we can rewrite the expression as:

an = (n - 1) - (n - 2) + (n - 3) - an - 3 + ... + (-1)^n-1(n - n) + (-1)^n

After simplifying and simplifying, we get:

an = 2n - 1 + (-1)^n / 4 + (-1)^n

Therefore, the correct option is (d) 2n - 1 + (-1)^n / 4 - (-1)^n.