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Chapter 7: Problem 28

Find the first term of an arithmetic sequence whose second term is 7 and whose fifth term is 11 .

Short Answer

Expert verified
The first term of the arithmetic sequence is \(\frac{17}{3}\).

Step by step solution

01

Define given variables and the equation for the arithmetic sequence.

We are given the following information about the arithmetic sequence: - Second term: 7 - Fifth term: 11 Let's denote the first term as \(a_1\), and the common difference as \(d\). The general form for an arithmetic sequence is: \[a_n = a_1 + (n-1)d\] where \(a_n\) is the nth term of the sequence.
02

Set up two equations for the second and fifth term.

Using the given information and the general form for an arithmetic sequence, let's create two equations for the second term and the fifth term. For the second term, where \(n = 2\): \[a_2 = a_1 + d = 7\] For the fifth term, where \(n = 5\): \[a_5 = a_1 + 4d = 11\]
03

Solve for common difference d.

We have two equations in terms of \(a_1\) and \(d\). Let's solve for \(d\) by subtracting the first equation from the second equation: \[(a_1 + 4d) - (a_1 + d) = 11 - 7\] Solving for \(d\): \[3d = 4\] \[d = \frac{4}{3}\]
04

Solve for the first term a_1.

With the common difference value, we can now find the first term. Using the first equation, we can plug in the value of \(d\) to solve for \(a_1\): \[a_1 + \frac{4}{3} = 7\] Subtracting \(\frac{4}{3}\) from both sides to solve for \(a_1\): \[a_1 = 7 - \frac{4}{3} = \frac{17}{3}\] So, the first term of the arithmetic sequence is \(\frac{17}{3}\).

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