usagirl007A

2021-08-13

Arithmetic or Geometric?

a) If${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence, is the sequence ${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ arithmetic?

b) If${a}_{1},{a}_{2},{a}_{3},\cdots$ is a geometric sequence, is the sequence $5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ geometric?

a) If

b) If

Khribechy

Skilled2021-08-14Added 100 answers

a) To show: the sequence ${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ is arithmetic.

Given:

The sequence${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence.

Approach:

In arithmetic sequence, difference between the terms remains constant throughout.

Calculation:

The sequence${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence.

${a}_{2}-{a}_{1}={a}_{3}-{a}_{2}=\cdots \cdots \left(1\right)$

$({a}_{2}+2)-({a}_{1}+2)=({a}_{2}-{a}_{1})$

$({a}_{3}+2)-({a}_{2}+2)=({a}_{3}-{a}_{2})$

From equation (1)

$({a}_{3}+2)-({a}_{2}+2)=({a}_{2}-{a}_{1})$

As the differences between the terms are same then it is an arithmetic sequence.

Conclusion: hence, the sequence${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ is arithmetic.

b) To show:$5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ is a geometric sequence.

Given: The sequence${a}_{1},{a}_{2},{a}_{3},\cdots$ is geometric sequence.

Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.

Calculation:

$\frac{{a}_{2}}{{a}_{1}}=\frac{{a}_{3}}{{a}_{2}}=\cdots \cdots \left(1\right)$

Multiplying equation (1) by 5

$\frac{5{a}_{2}}{5{a}_{1}}=\frac{5{a}_{3}}{5{a}_{2}}=\cdots$

Then it is a geometric sequence.

Conclusion: hence, the sequence$5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ is a geometric sequence.

Given:

The sequence

Approach:

In arithmetic sequence, difference between the terms remains constant throughout.

Calculation:

The sequence

From equation (1)

As the differences between the terms are same then it is an arithmetic sequence.

Conclusion: hence, the sequence

b) To show:

Given: The sequence

Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.

Calculation:

Multiplying equation (1) by 5

Then it is a geometric sequence.

Conclusion: hence, the sequence

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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