Pre calc help please!!

Question

1)     Use the following series to find the following below: 25 + 18 + 11 +....

a)      Find the sum of the first 22 terms in the series.

b)     Write the series in sigma notation for the first 22 terms in the series.

Answer 1

If we assume this is an arithmetic series with first term 25 and difference -7, then

then 22nd term is 25 +(21)(-7) = -122

The sum is 11(25 - 122) ....the average of the fist and last term * the number of terms

The sigma notation is ∑[25+(n-1)(-7)] for n=1 to 22 (I can't write the limits directly on the sigma!)

Answer 2

The generating sequence is arithmetic, with d = - 7 , and t₁ = 25. So the explicit formula for the nth term is

t_n = 25 - 7(n - 1) = -7n +32 (either is fine). We can use a formula for sum of the 1st n terms of an arith. series:

S_n = n/2(a₁ + a_n).

We need a₂₂ = -7(22) + 32 = - 122 to get. S₂₂ = 22/2(25 - 122) = -1,067

In sigma notation: Σ_n=1²² (-7n + 32) = -1,067