How could I solve $\sum_{k=3}^7 k^2-1$ by using the formula

Question

This afternoon I am dealin with with this question: $$\sum_{k=3}^7 k^2-1$$ When I using Microsoft Math solver for this question or just plug in every numbers, I get 130. However, when I using the formular equation, I got 140.
The formula I use is on page 420 on big ideas math algebra 2
The image
This is the work: $$ a_3=9-1=8 $$

$$ a_7=49-1=48 $$

$$ n\frac{a_3+a_n}{2} =5\frac{8+48}{2} =140 $$

I wonder is there anything I do wrong in the qustion. How could I fix it? What knowledge points I did not get?

Answer 1

I'll assume you mean $\sum_{k=3}^7 (k^2-1)$

The terms in the sequence are $3^2-1 = 8,4^2-1 = 15,5^2-1 = 24,6^2-1 = 35$ and $7^2-1 = 48$.

$8, 15, 24, 35, 48$ does not form an arithmetic sequence so that formula does not apply. $15-8 = 7$ and $24-15 = 9$, and since $7 \neq 9$, this is not an arithmetic sequence.

Answer 2

Suppose you are asking $\sum^7_{k=3}(k^2-1)$. $$ \begin{align*} \sum^7_{k=3}(k^2-1)&=\sum^7_{k=1}k^2-\sum^2_{k=1}k^2-\sum^7_{k=3}1 \\ &=\frac{7(7+1)(2\times 7+1)}{6}-\frac{2(2+1)(2\times 2+1)}{6}-5 \\ &=140 - 5 - 5\\ &=130 \end{align*} $$