Math Problem Solver

The sum of the sequence is $\displaystyle 20$

This is a geometric sequence.

The nth term in this sequence is given by the formula:

Explicit Formula: $$a_n=- 60 \left(-1\right)^{n} 2^{- n}$$

Recursive Formula: $$a_n=- \frac{a_{n-1}}{2}, \text{where } a_1=30$$

Summation Formula:

Option 1:

$$S_n=\frac{a_1 (r^n - 1)}{r - 1}$$ where $r\neq1$, $r$ is the common ratio, $a_1$ is the 1^st term, and $n$ is the is the term number.

Option 2:

$$\begin{aligned} S_n&=\sum_{i=1}^{n} a_{i} \\ &=\sum_{i=1}^{n} - 60 \left(-1\right)^{i} 2^{- i} \\ &=- 20 \left(-1\right)^{n} 2^{- n} + 20 \end{aligned}$$where $n$ is the is the term number.

This is an infinite sequence. Take the sum.$$\sum_{n=1}^{\infty} - 60 \left(-1\right)^{n} 2^{- n} = 20$$