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Sum of the first n terms of a PG


Be the PG (the1, a2, a3, a4,… , Theno ,…). For calculating the sum of the first n terms Sno, let's consider the following:
sno = a1 + a2 + a3 + a4 +… + An-1 + ano

Multiplying both members for the reason q we have:
sno.q = a1 . q + a2 .q +… + an-1 . q + ano .q

According to the definition of PG, we can rewrite the expression as:
sno . q = a2 + a3 +… + Ano + ano . what

Note that the2 + a3 +… + An is equal to Sno - a1 . So instead, comes:
sno . q = Sno - a1 + ano . what

Hence, simplifying conveniently, we will arrive at the following sum formula:

If we replace an = a1 . whatn-1, we will get a new presentation for the sum formula, namely:

Example:

Calculate the sum of the first 10 terms of PG (1,2,4,8,…)
We have:

Note that in this case the1 = 1.

Next: Sum of the Terms of an Infinite PG