Consider the sequence below : -2,1,6,13,22,... What explicit expression can be used to find the nth term of this sequence ?

Question

I really need help with this one

Answer 1

Looks like it's adding the next odd number to -2, starting by adding 3

a(1) = -2

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

a(2) = a(1) + 2(2)-1

a(3) = a(2) + 2(3)-1 = a(1)+2(2) -1 + 2(3)-1

:

n

a(n) = a(1) + ∑(2i-1)

i=2

Starting summation with 1 instead of 2 have to subtract 1

n

a(n) = -2 + ∑(2i-1) -1

i=1

= -3 + 2∑i - ∑1 = -3 +2n(n+1)/2 -n = -3 +n²+n -n = -3 +n²

a(n) = -3 +n²

Checking:

a(1) = -3 +1 = -2

a(2) = -3 +4 = 1

a(3) = -3 +9 = 6

a(4) = -3 +16 = 13

a(5) = -3 + 25 = 22

Answer 2

a_n = -2 + (2n - 1)