Arithmetic Series: find the sum of 25 terms given 2 terms and their values

Question

I need help with this question:

"Find S₂₅, given an arithmetic series whose 8th term is 16 and whose 13th term is 81."

What I did was:

1. Found the common difference (d) like this:

81 - 16 = 65
13 - 8 = 5
65 ÷ 5 = 13 (common difference)

2.Found the 1st term like this:

16 - 8 x 13 = -88 (T₁)

3. Substituted it into the formula S_n = n/2(2a + d(n - 1)), where n = 25,
a = T₁ = -88, and d = 13:

S₂₅ = 25/2 ( 2(-88) + 13(25-1)) = 1700 (the sum of 25 terms)

But the answer to this question in my book is S₂₅ = 2025

Please explain me where is my mistake or what I am doing wrong. Thank you.

Answer 1

$$d=\frac{a_n-a_m}{n-m}$$ for all $n\neq m$.

Thus, $$d=\frac{16-81}{8-13}$$ or $$d=13.$$ Now, $$a_n=a_1+(n-1)d,$$ which gives $$16=a_1+7\cdot13$$ or $$a_1=-75.$$ Here is your mistake.