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Concept explainers
(a)
To explain:
The sequence and the way to denote the terms of a sequence.
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Explanation of Solution
A sequence is a function a whose domain is the set of natural numbers. The terms of the sequence are the function values.
The terms of the sequence can be expressed as,
a1, a2, a3,…,an,…
Here, the first term is a1, second term is a2 and the general term an is called the nth term.
The sequence with nth term can be written as an.
(b)
To find:
The formula for sequence of even numbers and formula for sequence of odd numbers.
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Answer to Problem 1CC
The formula for sequence of even numbers and formula for sequence of odd numbers are an=2n and an=2n−1 respectively.
Explanation of Solution
Calculation:
The sequence of even number is,
2, 4, 6, 8 , ⋯
The first term of the sequence is 2, the second term of the sequence is 4, the third term is 6 and common difference is calculated as,
d=a2−a1=4−2=2
Substitute 2 for a and 2 for d the formula of the sequence of even numbers can be expressed as,
an=2+(n−1)(2)=2+2n−2=2n
The sequence of odd number is,
1, 3, 5, 7, ⋯
The first term of the sequence is 1, the second term of the sequence is 3, the third term is 5 and common difference is calculated as,
d=a2−a1=3−1=2
Substitute 2 for a and 2 for d the formula of the sequence of odd numbers can be expressed as,
an=1+(n−1)(2)=1+2n−2=2n−1
Conclusion:
Hence, the formula for sequence of even numbers and formula for sequence of odd numbers are an=2n and an=2n−1 respectively.
(c)
To find:
The first three terms and the 10thterm of the sequence given by an=nn+1.
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Answer to Problem 1CC
The first three terms of the sequence an=nn+1 are 12, 23, 34 and the 10thterm is 1011.
Explanation of Solution
Given information:
The sequence is,
an=nn+1 .............(1)
Calculation:
Substitute 1 for n in equation (1) to obtain the first term.
a1=11+1=12
Substitute 2 for n in equation (1) to obtain the second term.
a2=22+1=23
Substitute 3 for n in equation (1) to obtain the third term.
a3=33+1=34
Substitute 10 for n in equation (1) to obtain the tenth term.
a10=1010+1=1011
Conclusion:
Hence, the first three terms of the sequence an=nn+1 are 12, 23, 34 and the 10thterm is 1011.
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Chapter 8 Solutions
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