Question

Derive the formula for calculating the arithmetic series : Sn=n/2(2a+(n-1)d)

Answer 1

IDK man thats alot of numbers

Answer 2

think about it. If the series has n terms, then
it can be divided up into pairs, starting from the ends.
Each pair adds up to
a1 + an = a + a+(n-1)d = 2a + (n-1)d
a2 + a(n-1) = a+d + a+(n-2)d = 2a + (n-1)d
and so on. There are n/2 of these pairs, so you get the formula

more formally,
Sn = a1 + a2 + a3 ... + an
Sn = a1 + a1+d + a1+2d + ... + a1 + (n-1)d
Sn = an + an-d + an-2d + ... + an-(n-1)d
Now add those last two lines:
2Sn = a1+1n + a1+1n + a1+an + ... a1+an
so Sn = 1/2 * n(2a1+an) = n/2 (2a1+(n-1)d)

You can also use induction. Give it a try.