Question

Find s25 for 3 + 7 + 11 + 15

Answer 1

The common difference is d = 4 because we add 4 to each term to get the next one.

The starting term is a1 = 3

The nth term of this arithmetic sequence is 
an = a1 + d(n-1)
an = 3 + 4(n-1)
an = 3 + 4n-4
an = 4n - 1

Plug in n = 25 to find the 25th term
an = 4n - 1
a25 = 4*25 - 1
a25 = 100 - 1
a25 = 99

So we're summing the series : 3+7+11+15+...+99

We could write out all the terms and add them all up. That's a lot more work than needed though. Luckily we have a handy formula to make things a lot better
The sum of the first n terms is Sn. The formula for Sn is
Sn = n*(a1+an)/2

Plug in n = 25 to get
Sn = n*(a1+an)/2
S25 = 25*(a1+a25)/2

Then plug in a1 = 3 and a25 = 99. Then compute to simplify

S25 = 25*(a1+a25)/2
S25 = 25*(3+99)/2
S25 = 25*(102)/2
S25 = 2550/2
S25 = 1275

The final answer is 1275