# Finding the Sum of Geometric Series of n Terms being Inserted between 1 and 2

Suppose $n$ consecutive geometric terms are inserted between $1$ and $2$. Write the sum of these $n$ terms in terms of $n$.

$$\Large \underbrace{1, \overbrace{u_1, u_2, u_3, \cdots, u_{n}}^{S_n}, 2}_{S_{n+2}}$$

\require{AMSsymbols} \displaystyle \begin{align} S_{n+2} &= 1+u_1 + u_2 + u_3 + \cdots + u_n + 2 \\ &= 1 + S_n + 2 \\ S_n &= S_{n+2}-3 \color{red}{\cdots (1)} \\ S_{n+2} &= \frac{a(r^{n+2}-1)}{r-1} &\color{green}{a \text{ is the first term, and } r \text{ is the common ratio}} \\ &= \frac{r^{n+2}-1}{r-1} \color{red}{\cdots (2)} &\color{green}{a=1} \\ u_1 &= r &\color{green}{1 \times r = u_1} \\ u_2 &= r^2 \\ u_3 &= r^3 \\ &\cdots \\ u_n &= r^{n} \\ 2 &= r^{n+1} &\color{green}{u_n \times r = 2} \\ r &= 2^{\frac{1}{n+1}} \\ S_{n+2} &= \frac{\left(2^{\frac{1}{n+1}}\right)^{n+2}-1}{2^{\frac{1}{n+1}}-1} &\color{red}{\text{by } (2)} \\ &= \frac{2^{\frac{n+2}{n+1}}-1}{2^{\frac{1}{n+1}}-1} \\ \therefore S_n &= \frac{2^{\frac{n+2}{n+1}}-1}{2^{\frac{1}{n+1}}-1}-3 &\color{red}{\text{by } (1)} \end{align} ## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## Induction Made Simple: The Ultimate Guide

“Induction Made Simple: The Ultimate Guide” is your gateway to mastering the art of mathematical induction, demystifying a powerful tool in mathematics. This ultimate guide…

## Simplified Method for Calculating First Term and Common Ratio in Infinite Geometric Series

Hello, math enthusiasts! Today, we’re delving into the captivating realm of infinite geometric series and discovering how to easily determine the “First Term” and the…

## High School Math for Life: Making Sense of Earnings

Salary Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis,…

## Everything You Need to Know about Arithmetic Sequences

An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same fixed number. It can also be referred…