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Chapter Analysis
Intermediate23 pages • EnglishQuick Summary
This chapter on Arithmetic Progressions introduces students to the concept of numbers arranged according to a fixed pattern, where each successive term is obtained by adding a constant called the common difference. It covers the general terms, sum of n terms, and practical applications such as calculating salaries and savings. The chapter also includes exercises and examples to help students learn how to determine the terms and sums of an AP.
Key Topics
- •Definition of an Arithmetic Progression
- •Common difference in APs
- •General term of an AP
- •Sum of the first n terms
- •Applications of AP in daily life
- •Determining terms in an AP
- •Finite and infinite APs
- •Solving practical problems using AP
Learning Objectives
- ✓Understand the concept of an arithmetic progression
- ✓Identify the common difference in a sequence
- ✓Calculate the nth term of an AP
- ✓Compute the sum of n terms of an AP
- ✓Apply knowledge of APs to solve real-life problems
- ✓Recognize and create both finite and infinite APs
Questions in Chapter
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
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Write first four terms of the AP, when the first term a and the common difference d are given as follows: (i) a = 10, d = 10 (ii) a = –2, d = 0
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For the following APs, write the first term and the common difference: (i) 3, 1, – 1, – 3, . . . (ii) – 5, – 1, 3, 7, . . .
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Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
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Fill in the blanks in the following table, given that a is the first term, d the common difference and a_n the nth term of the AP
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Additional Practice Questions
Determine the common difference if the third and seventh terms of an AP are 11 and 23, respectively.
mediumAnswer: Using the formula for the nth term: a_3 = a + 2d = 11 and a_7 = a + 6d = 23. Solving, we find d = 3.
If the first term of an AP is 5 and the last term is 95 with the sum being 500, find the number of terms.
easyAnswer: Using the sum formula S = n/2 * (a + l) => 500 = n/2 * (5 + 95). Solving, n = 10.
An AP consists of 30 terms in which the last term is 60. If the common difference is 2, find the first term.
mediumAnswer: Using the nth term formula a_n = a + (n-1)d, we have 60 = a + 29*2. Solving, a = 2.
Is the sequence 4, 7, 10, 13,... an AP? Justify your answer.
easyAnswer: Yes, the sequence is an AP with common difference d = 3 (7-4 = 10-7 = 13-10 = 3).
Calculate the sum of the first 15 terms of an AP with first term 2 and common difference 4.
hardAnswer: Using S_n = n/2 * (2a + (n-1)d), we have S_15 = 15/2 * (2*2 + 14*4), resulting in 480.