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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
In this chapter, sequences and series are introduced as fundamental mathematical concepts. It covers the understanding of arithmetic and geometric progressions, along with the arithmetic mean, geometric mean, and the relationship between them. The chapter provides tools to find the nth term and the sum of the first n terms in both arithmetic and geometric sequences. It also touches upon special series like sum squares and cubes of natural numbers.
Key Topics
- •Sequences and Series
- •Arithmetic Progression
- •Geometric Progression
- •Arithmetic Mean
- •Geometric Mean
- •Sum of Series
- •Special Series
- •nth Term Calculation
Learning Objectives
- ✓Understand the definition of sequences and series
- ✓Differentiate between arithmetic and geometric progressions
- ✓Calculate the nth term in progressions
- ✓Find the sum of n terms in a sequence
- ✓Explore the relationship between arithmetic and geometric means
- ✓Use sequences and series to solve practical problems
Questions in Chapter
Find the 20th and nth terms of the G.P. 5/2, 5/4, 5/8, ...
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For what values of x, the numbers -2/7, x, -7/2 are in G.P.?
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The sum of first three terms of a G.P. is 13/12 and their product is – 1. Find the common ratio and the terms.
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Additional Practice Questions
If the sum of the first n terms of an arithmetic progression is 150 and the first term is 5, find the common difference.
mediumAnswer: Let the sum be S_n = n/2 * (2a + (n-1)d). Here S_n = 150, a = 5. Solving 150 = n/2 * (10 + (n-1)d) will give d.
A company's sales in the first quarter is $1,000, and then each quarter's sales increase by 10% of the previous quarter. What are the sales at the end of the third quarter?
easyAnswer: The sales form a geometric sequence with first term a = 1000 and common ratio r = 1.1. The third term is a * r^2 = 1000 * 1.1^2 = $1,210.
A sequence is defined by the recursive formula a_n = 2a_{n-1} + 3. If a_1 = 1, find the fourth term, a_4.
mediumAnswer: Using a_1 = 1, a_2 = 2(1) + 3 = 5, a_3 = 2(5) + 3 = 13, a_4 = 2(13) + 3 = 29.
Prove that the sum of cubes of first n natural numbers is (n(n + 1) / 2)^2.
hardAnswer: Use mathematical induction. Base case n=1, 1^3 = 1 = (1(1+1)/2)^2 holds. Assume it's true for n=k, show it's true for n=k+1.
If the geometric mean of two numbers is 6 and their sum is 18, find the numbers.
easyAnswer: Let numbers be a and b. Given sqrt(ab) = 6 and a + b = 18. So, ab = 36. Solving, (a,b) = (12,6) or (6,12).