Question

Find the sum of the first 25 terms of the arithmetic sequence -2,2,6,10,14?

Answer 1

`1150`

Explanation:

We have the formula to tell the sum of the first `n` numbers:

`\sum_{i=1}^n i = \frac{n(n+1)}{2}`

You're not summing the first `25` numbers, but a translated version: your series is `a_n = 4n-6`, and you want to compute

`\sum_{i=1}^{25} a_i = \sum_{i=1}^{25} 4i-6`

we can rerwite the sum as

`\sum_{i=1}^{25} 4i-\sum_{i=1}^{25}6 = 4\sum_{i=1}^{25} i-\sum_{i=1}^{25}6`

And we're ready to answer: the first term is actually the sum of the first `25` numbers, multiplied by 4. Therefore, we have

`4\sum_{i=1}^{25} i = 4\frac{25\cdot26}{2} = 2\cdot25\cdot26=1300`

The second term is even easier: we have to sum `6` with itself `25` times...this is called multiplication :)

`\sum_{i=1}^{25}6 = 6\cdot 25 = 150`

So, the answer is `1300-150=1150`