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Answer :
To solve the problem step-by-step, we need to address each part of the question clearly:
1. Understanding the sequence and progression types:
- The sequence given is `1, -2, 4, -8, ...`, and this follows a geometric progression (G.P.) because each term is obtained by multiplying the previous term by a constant factor.
- The common differences given as `1/2, 1, 3/2, ...` appear to be talking about an arithmetic sequence in another context.
- However, we are focusing on the arithmetic progression (A.P.) part mentioned later.
2. Arithmetic Progression (A.P.) Calculation:
- We are given an arithmetic progression with a first term of 1 and a common difference of 3.
- To find the 8th term of this A.P., we use the formula for the nth term of an arithmetic progression:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Finding the 8th term:
- Plug in the values: [tex]\(a_1 = 1\)[/tex], [tex]\(d = 3\)[/tex], and [tex]\(n = 8\)[/tex]:
[tex]\[
a_8 = 1 + (8 - 1) \cdot 3 = 1 + 21 = 22
\][/tex]
4. The effect of adding 2 to every 6th term:
- The question states that 2 is added to every 6th term of the progression.
- This action does not affect the common difference between consecutive terms.
- The common difference remains 3.
So, summarizing the solution:
- The 8th term of the arithmetic progression is 22.
- The common difference of the arithmetic progression remains 3 even after the adjustment.
1. Understanding the sequence and progression types:
- The sequence given is `1, -2, 4, -8, ...`, and this follows a geometric progression (G.P.) because each term is obtained by multiplying the previous term by a constant factor.
- The common differences given as `1/2, 1, 3/2, ...` appear to be talking about an arithmetic sequence in another context.
- However, we are focusing on the arithmetic progression (A.P.) part mentioned later.
2. Arithmetic Progression (A.P.) Calculation:
- We are given an arithmetic progression with a first term of 1 and a common difference of 3.
- To find the 8th term of this A.P., we use the formula for the nth term of an arithmetic progression:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Finding the 8th term:
- Plug in the values: [tex]\(a_1 = 1\)[/tex], [tex]\(d = 3\)[/tex], and [tex]\(n = 8\)[/tex]:
[tex]\[
a_8 = 1 + (8 - 1) \cdot 3 = 1 + 21 = 22
\][/tex]
4. The effect of adding 2 to every 6th term:
- The question states that 2 is added to every 6th term of the progression.
- This action does not affect the common difference between consecutive terms.
- The common difference remains 3.
So, summarizing the solution:
- The 8th term of the arithmetic progression is 22.
- The common difference of the arithmetic progression remains 3 even after the adjustment.
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