Sums of Geometric Sequences

Key Questions

The formula is: ${S}_{n} = \frac{{a}_{1} \left(1 - {q}^{n}\right)}{1 - q}$

Explanation:

To proove this formula you have first to write the sum of a geometric sequence:

${S}_{n} = {a}_{1} + {a}_{1} \cdot q + {a}_{1} \cdot {q}^{2} + \ldots {a}_{1} \cdot {q}^{n - 1}$

${S}_{n} = {a}_{1} \cdot \left(1 + q + {q}^{2} + \ldots {q}^{n}\right)$

We can multiply the last equality by $\left(1 - q\right)$
We get:

$\left(1 - q\right) \cdot {S}_{n} = {a}_{1} \cdot \left(1 - q\right) \cdot \left(1 + q + {q}^{2} + \ldots {q}^{n}\right)$

The right hand side can be written as ${a}_{1} \cdot \left({1}^{n} - {q}^{n}\right)$ or
${a}_{1} \cdot \left(1 - {q}^{n}\right)$. So finally we get:

$\left(1 - q\right) \cdot {S}_{n} = {a}_{1} \cdot \left(1 - {q}^{n}\right)$

Supposing $q \ne 1$, we can divide both sides by $1 - q$ and get:

${S}_{n} = \frac{{a}_{1} \left(1 - {q}^{n}\right)}{1 - q}$

That concludes the proof.

There can be many problems. I will just give a few.

Explanation:

Examples:

1)

A bank offers an account with interest rate 10 % per annum. A client wants to put $10,000 on such account for 5 years. How much money will he get after 5 years. 2) A man runs 10 meters in the first second, in every other second he runs twice as far as is in previous one. How long will he run in 10 seconds? (In this example you could also ask what time will he get on a distance of 100m ?) 3) In this example you can combine arithmetic and geometric sequence: A student wants to take a holiday job for 14 days. He can choose from 2 offers: 1st: He earns$50 on first day and every other day he earns $5 more than in a previous day . 2nd He gets$10 on first day and every other day he earns twice as much as on a previous day.
Which offer is better for him?

4)
A plant is 1m tall at the beginning. Every day it grows twice as tall as it was the day before. In which day will it exceed 20m?