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 รท 5 = 13 (common difference)
2.Found the 1st term like this:
16 – 8 x 13 = -88 (T1)
3. Substituted it into the formula Sn = 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.
Best Answer
$$d=\frac{a_n-a_m}{n-m}$$ 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.