/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Find the indicated sum. Use the ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 31

Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{8} 3^{i}$$

Short Answer

Expert verified
The sum of the geometric sequence for the first 8 terms is 19683.

Step by step solution

01

Identifying the Parameters

In the geometric series \(\sum_{i=1}^{8} 3^{i}\), the first term \(a = 3^1 = 3\), the common ratio \(m = 3\), and the number of terms \(n = 8\).
02

Using the Sum of Geometric Sequence Formula

We know that the formula for the sum of the first \(n\) terms of a geometric sequence is \(S_n = a(m^n - 1) / (m - 1)\) where \(a\) is the first term, \(m\) is the common ratio and \(n\) is the number of terms. Substituting the values found in Step 1 into the equation results in \(S_8 = 3(3^8 - 1) / (3 - 1)\).
03

Calculating the Sum

Now, we just have a simple evaluation problem. After calculating, we see that \(S_8 = 3(3^8 - 1) / (3 - 1) = 19683\).

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Most popular questions from this chapter

Which one of the following is true? a. The binomial expansion for \((a+b)^{n}\) contains \(n\) terms. b. The Binomial Theorem can be written in condensed form as \((a+b)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l}n \\ r\end{array}\right) a^{n-r} b^{r}\). c. The sum of the binomial coefficients in \((a+b)^{n}\) cannot be \(2^{n}\). d. There are no values of \(a\) and \(b\) such that \((a+b)^{4}=a^{4}+b^{4}\)

Explain how to evaluate \(\left(\begin{array}{l}n \\ r\end{array}\right) .\) Provide an example with your explanation.

If \(f(x)=x^{5},\) find \(\frac{f(x+h)-f(x)}{h}\) and simplify.

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(y^{3}-1\right)^{21} $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+4)^{3} $$
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