Partial sums of infinite geometric series.

Find the sum of a geometric series (implies the whole infinite series).

{ 2, 2/5, 2/25, . . . }

becomes 2 + 2/5 + 2/25 + 2/125

The formula of a partial sum is:

Sn = t1 / 1 - r; r = 1/5 and t1 = 2
2 / 1 - 1/5
= 2 / (4/5)
= 2 * 5/4
= 5 / 2
= 2.5

e.x. An infinite convergent geometric sequence has as its infinite sum, 16 with a common ratio 1/2. What is the first term in the sequence?

S = 16; r= 1/2

S = t1 / 1 - r
16 = t1 / 1 - 1/2
16 = t1 / (1/2)
8 = t1

This reminds me of Zeno's paradox: if an archer shoots an arrow at a target, it'll travel 1/2 way to its target. So, for the first round, it's 1/2 from the target, 1/4 from the target, 1/8, 1 /16, 1 / 36, 1 / 72 . . .

So, it never reaches its target. Yet real world experimentation disproves this. All this proves is that if we break a movement into smaller steps, the movement becomes impossible. That's kinda the same with these infinite geometric series. It keeps adding up and up and up infinity. Yet it reaches a certain number. Interesting. . . .