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Chapter 22: Chapter 22

Problem 655

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If the 6 th term of an arithmetic progression is 8 and the \(11^{\text {th }}\) term is \(-2\), what is the 1 st term? What is the common difference?

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The first term of the arithmetic progression is 18 and the common difference is -2.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the equations for the 6th and 11th terms

Using the formula of an arithmetic progression, we write the equations for the 6th term and the 11th term: 6th term: \(8 = a + (6-1)d\) 11th term: \(-2 = a + (11-1)d\)

Step 2: Solve for the common difference (d)

Let's use two variables for our ease: x = a + 5d (6th term equation) y = a + 10d (11th term equation) Subtract the first equation from the second equation to eliminate "a": (y - x) = (10d - 5d) -2 - 8 = 5d -10 = 5d d = -2

Step 3: Solve for the first term "a"

Now, we can substitute the value of "d" into either equation to solve for "a": 8 = a + 5(-2) 8 = a - 10 a = 18 So, the first term of the arithmetic progression is 18, and the common difference is -2.

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