Find the sum of a finite arithmetic sequence from n = 1 to n = 15, using the expression 2n+ 5.

Solution:

\(s_{n}=\sum_{n=1}^{15}(2n + 5)\)

We have to calculate the sum of given expression that is in an arithmetic sequence.

Consider the given expression,

\(s_{n}=\sum_{n=1}^{15}(2n + 5)\)

Apply sum rule, \(\sum (a_{n}+b_{n}) =\sum (a_{n})+ \sum (b_{n})\)

So \(s_{n}=\sum_{n=1}^{15}(2n + 5)\)

Using constant multiplication rule\(\sum c(a_{n}) = c\sum (a_{n})\)

\(s_{n}=2\sum_{n=1}^{15}(n)+ 5\sum_{n=1}^{15}(1)\)

Now first consider \(2\sum_{n=1}^{15}(n)\)

Using the sum of even numbers  formula,

\( \sum_{n=1}^{15}(2n)\) = [n(n+1)]

⇒ \(\sum_{n=1}^{15}2n \)= 15(15+1) = 15 × 16 = 240

Now consider \(5\sum_{n=1}^{15}(1)\)

\(\sum_{n=1}^{15}(1)\)

= 5.(15) = 75

\(s_{n}=\sum_{n=1}^{15}(2n + 5) = 2\sum_{n=1}^{15}(n) + 5\sum_{n=1}^{15}(1)\)

= 240 + 75

\(s_{n}\) = 315

Thus the sum of the finite sequence from n = 1 to 15 using the expression 2n + 5 = 315

---

Find the sum of a finite arithmetic sequence from n = 1 to n = 15, using the expression 2n+ 5.

Summary:

The sum of a finite arithmetic sequence from n = 1 to n = 15, using the expression 2n+ 5 is 345