Chapter 7: Q. 7 (page 591)

Give the first five terms of the following recursively defined sequence:
$a_{1} = 1$ , and $a_{k} = a_{k - 1} + 2$ for $k \geq 2$ .
Also, give a closed formula for the sequence.

Short Answer

Expert verified

${1, 3, 5, 7, 9} a_{k} = 2 k - 1; k \geq 1$

Step by step solution

Step1. Given Information

$Consider the sequence a_{k} = a_{k - 1} + 2, a_{1} = 1, k \geq 2$

$The objective is to find the first five terms of the sequence. Also, find the closed formula.$

$To find the terms of the sequence, substitute k = 2, 3, 4, 5 in a_{k} = a_{k - 1} + 2$

Step2. Substitution

$a_{2} = a_{2 - 1} + 2$

$= a_{1} + 2$ (using $a = 1$ )

$= 1 + 2$

Therefore, $a_{2} = 3$

Step3. Substitution

Substituting $k = 3$ ,

$a_{3} = a_{3 - 1} + 2$

$= a_{2} + 2$ using( $a_{2} = 3$ )

$= 3 + 2$

$= 5$

Therefore, $a_{3} = 5$

Step4. Substitution

Substituting $k = 4$ ,

$a_{4} = a_{4 - 1} + 2$

$= a_{3} + 2 u \sin g (a_{3} = 5) = 5 + 2 = 7$

Therefore, $a_{4} = 7$

Step5. Substitution

Substituting $k = 5$

$a_{5} = a_{5 - 1} + 2 = a_{4} + 2 u \sin g (a_{4} = 7) = 7 + 2 = 9$

Therefore, $a_{5} = 9$

Step6. Answer

Hence, the five terms of the sequence are ${1, 3, 5, 7, 9}$ .

Clearly, the given sequence is sequence of consecutive odd numbers.

$Hence, the closed formula is a_{k} = 2 k - 1, where k is the positive integer and k \geq 1 .$

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Give the first five terms of the following recursively defined sequence:
$a_{1} = 1$ , and $a_{k} = a_{k - 1} + 2$ for $k \geq 2$ .
Also, give a closed formula for the sequence.

Short Answer

Step by step solution

Step1. Given Information

Step2. Substitution

Step3. Substitution

Step4. Substitution

Step5. Substitution

Step6. Answer

One App. One Place for Learning.

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Give the first five terms of the following recursively defined sequence:a1=1, and ak=ak-1+2for k≥2. Also, give a closed formula for the sequence.

Short Answer

Step by step solution

Step1. Given Information

Step2. Substitution

Step3. Substitution

Step4. Substitution

Step5. Substitution

Step6. Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Geometry

Pure Maths

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Give the first five terms of the following recursively defined sequence:
$a_{1} = 1$ , and $a_{k} = a_{k - 1} + 2$ for $k \geq 2$ .
Also, give a closed formula for the sequence.