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Chapter 7: Problem 29

Find the \(100^{\text {th }}\) term of an arithmetic sequence whose tenth term is 5 and whose eleventh term is 8 .

Short Answer

Expert verified
The 100th term of the given arithmetic sequence is 275.

Step by step solution

01

Identify the common difference

As we know, the difference between consecutive terms is constant in an arithmetic sequence. We can find the common difference by subtracting the 10th term from the 11th term: \(8 - 5 = 3\). So, the common difference (d) is 3.
02

Write the general formula for the arithmetic sequence

To find the specific term of an arithmetic sequence, we can use the following formula: \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, n is the number of terms, and d is the common difference. In this case, we are looking for the 100th term. First, we need to find the first term (\(a_1\)).
03

Find the first term

We have the 10th term (\(a_{10}\)) and the common difference (d). We can use this information to find the first term (\(a_1\)). Using the formula from Step 2: \(a_{10} = a_1 + (10 - 1)d\) Substitute the given values: \(5 = a_1 + (9)(3)\) Solve for \(a_1\): \(a_1 = 5 - (9)(3) = 5 - 27 = -22\)
04

Find the 100th term using the formula

Now that we have \(a_1\) and the common difference (d), we can find the 100th term (\(a_{100}\)). Using the formula from Step 2: \(a_{100} = a_1 + (100 - 1)d\) Substitute the values: \(a_{100} = (-22) + (99)(3)\) Calculate the solution: \(a_{100} = -22 + (297) = 275\)
05

State the final answer

The 100th term of the given arithmetic sequence is 275.

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