Chapter 22: Chapter 22

Problem 660

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Given that the first term of an arithmetic sequence is 56 and the seventeenth term is 32 , find the tenth term and the twenty - fifth term.

Short Answer

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The tenth term of the arithmetic sequence is \(\dfrac{85}{2}\), and the twenty-fifth term is 20.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Define the arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is given by: \(a_n = a_1 + (n - 1)d\) where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference between consecutive terms.

Step 2: Calculate the common difference

We are given the first term (\(a_1 = 56\)) and the seventeenth term (\(a_{17} = 32\)). We can use the arithmetic sequence formula to find the common difference (d): \(a_{17} = a_1 + (17 - 1)d\) Substitute the given values into the formula: \(32 = 56 + (16)d\) Now, we solve for d: \(16d = 32 - 56\) \(16d = -24\) \(d = -\dfrac{24}{16} = -\dfrac{3}{2}\) The common difference, d, is \(-\dfrac{3}{2}\)

Step 3: Find the tenth term

Now that we have the common difference, we can find the tenth term, \(a_{10}\), using the arithmetic sequence formula: \(a_{10} = a_1 + (10 - 1)d\) Substitute the values: \(a_{10} = 56 + (9)(-\dfrac{3}{2})\) \(a_{10} = 56 - \dfrac{27}{2}\) \(a_{10} = \dfrac{85}{2}\) The tenth term, \(a_{10}\), is \(\dfrac{85}{2}\).

Step 4: Find the twenty-fifth term

Similarly, we can find the twenty-fifth term, \(a_{25}\), using the arithmetic sequence formula: \(a_{25} = a_1 + (25 - 1)d\) Substitute the values: \(a_{25} = 56 + (24)(-\dfrac{3}{2})\) \(a_{25} = 56 - 36\) \(a_{25} = 20\) The twenty-fifth term, \(a_{25}\), is 20. To summarize, the tenth term of the arithmetic sequence is \(\dfrac{85}{2}\), and the twenty-fifth term is 20.

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Chapter 22: Chapter 22

Given that the first term of an arithmetic sequence is 56 and the seventeenth term is 32 , find the tenth term and the twenty - fifth term.

Short Answer

Step by step solution

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Step 1: Define the arithmetic sequence formula

Step 2: Calculate the common difference

Step 3: Find the tenth term

Step 4: Find the twenty-fifth term

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More chapters from the book ‘Algebra and Trigonometry’

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Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 23

Chapter 24

Chapter 25

Chapter 26

Chapter 27

Chapter 28

Chapter 29

Chapter 30

Chapter 31

Chapter 32

Chapter 33

Chapter 34

Chapter 35

Chapter 36

Chapter 37

Chapter 38

Chapter 39

Chapter 40

Chapter 41

Chapter 42

Chapter 43

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Chapter 46

Chapter 47

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