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Chapter 13: Problem 23

Find the indicated term for each geometric sequence $$ a_{1}=2, r=5 ; \quad a_{10} $$

Short Answer

Expert verified
The 10th term is 3,906,250.

Step by step solution

01

- Recall the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is given by \[a_n = a_1 \times r^{(n-1)}\]where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
02

- Identify given values

From the problem, we know that: \(a_1 = 2\), \(r = 5\), and we need to find \(a_{10}\).
03

- Plug in the values

Substitute \(a_1 = 2\), \(r = 5\), and \(n = 10\) into the formula:\[ a_{10} = 2 \times 5^{(10-1)}\]
04

- Simplify the exponent

Calculate the exponent:\[5^{(10-1)} = 5^9\]
05

- Calculate the power of the common ratio

Now calculate \(5^9\):\[ 5^9 = 1953125\]
06

- Complete the multiplication

Substitute back to get the result:\[a_{10} = 2 \times 1953125 = 3906250\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

nth term of geometric sequence
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. To find the nth term of a geometric sequence, we use the formula: \[a_n = a_1 \times r^{(n-1)}\] Here,
  • \(a_n\) is the nth term we want to find
  • \(a_1\) is the first term of the sequence
  • \(r\) is the common ratio
  • \(n\) is the term number
Using this formula, we can find any term in the sequence if we know the first term and the common ratio.

In the given exercise, you find the 10th term, and the values are: \(a_1 = 2\), \(r = 5\), and \(n = 10\). You just substitute these into the formula.
common ratio
The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It's denoted by \(r\).
  • If the common ratio \(r\) is greater than 1, the sequence grows exponentially.
  • If \(r\) is between 0 and 1, the sequence decreases and approaches zero.
  • If \(r\) is negative, the terms of the sequence will alternate in sign.

In our example, the common ratio \(r = 5\), which means each term is 5 times the previous one.
exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In a geometric sequence formula, exponentiation is used to raise the common ratio to the power of \(n-1\).

In our exercise:
  • We start with the common ratio \(r = 5\).
  • To find the 10th term, we calculate \(5^{(10-1)}\) which simplifies to \(5^9\).
  • Exponentiation tells us to multiply the base (5) by itself the number of times indicated by the exponent (9).

Thus, \(5^9 = 1953125\). Finally, multiplying by the first term, we get \(a_{10} = 2 \times 1953125 = 3906250\).

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