Chapter 22: Chapter 22

Problem 661

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The 54 th and 4 th terms of an arithmetic progression are 61 and 64 ; find the \(23 \mathrm{rd}\) term.

Short Answer

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The 23rd term of the arithmetic progression is \(\frac{3143}{50}\).

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Understand the formula for arithmetic progressions

For an arithmetic progression, the formula to find the nth term is: \(a_n = a_1 + (n-1)d\), where: - \(a_n\) is the nth term of the progression - \(a_1\) is the first term of the progression - n is the position of the term in the progression - d is the common difference between consecutive terms in the progression

Step 2: Write the given information

We are given the 54th term and the 4th term. So, we have: \(a_{54} = 61\) \(a_{4} = 64\)

Step 3: Write the formulas for the given terms

Using the formula from Step 1, we can write the formulas for the given terms: \(a_{54} = a_1 + 53d = 61\) \(a_{4} = a_1+3d = 64\)

Step 4: Solve for the common difference d

Subtract the second equation from the first equation to eliminate \(a_1\): \((a_1 + 53d) - (a_1+3d) = 61 - 64\) This simplifies to: \(50d = -3\) Now, divide by 50: \(d = -\frac{3}{50}\)

Step 5: Solve for the first term a1

Plug the value of d into one of the equations (here we use the equation for \(a_{4}\)): \(a_1 + 3\left(-\frac{3}{50}\right) = 64\) Multiply out the term: \(a_1 - \frac{9}{50} = 64\) Add the fraction to the other side: \(a_1 = 64 + \frac{9}{50}\) Convert 64 to a fraction with the same denominator: \(a_1 = \frac{3200}{50} + \frac{9}{50}\) Add the fractions together: \(a_1 = \frac{3209}{50}\)

Step 6: Find the 23rd term a23

Now that we have the common difference and the first term, we can find the 23rd term using the formula from Step 1: \(a_{23} = a_1 + 22d\) Plug in the values of \(a_1\) and d: \(a_{23} = \frac{3209}{50} + 22\left(-\frac{3}{50}\right)\) Multiply out the term: \(a_{23} = \frac{3209}{50} - \frac{66}{50}\) Combine the fractions: \(a_{23} = \frac{3209-66}{50}\) Simplify: \(a_{23} = \frac{3143}{50}\) The 23rd term of the arithmetic progression is \(\frac{3143}{50}\).

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Chapter 22: Chapter 22

The 54 th and 4 th terms of an arithmetic progression are 61 and 64 ; find the \(23 \mathrm{rd}\) term.

Short Answer

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Step 1: Understand the formula for arithmetic progressions

Step 2: Write the given information

Step 3: Write the formulas for the given terms

Step 4: Solve for the common difference d

Step 5: Solve for the first term a1

Step 6: Find the 23rd term a23

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Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

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Chapter 12

Chapter 13

Chapter 14

Chapter 15

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Chapter 18

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Chapter 23

Chapter 24

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Chapter 26

Chapter 27

Chapter 28

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Chapter 33

Chapter 34

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Chapter 38

Chapter 39

Chapter 40

Chapter 41

Chapter 42

Chapter 43

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