Answer :
To solve this problem, we need to find the sum of the first 69 terms of an arithmetic progression (A-P) where the 35th term is given as 69.
Here's a step-by-step explanation:
1. Understand the Terms and Definitions:
- The 35th term of the A-P is 69.
- We need to find the sum of the first 69 terms.
2. Use the Formula for the nth Term of an A-P:
- The formula for the nth term ([tex]\(a_n\)[/tex]) of an arithmetic progression is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.
3. Apply the Given Information:
- From the information, the 35th term ([tex]\(a_{35}\)[/tex]) is 69, so:
[tex]\[
a_{35} = a_1 + 34d = 69
\][/tex]
4. Find the Sum of the First 69 Terms:
- The formula for the sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) of an A-P is:
[tex]\[
S_n = \frac{n}{2} \left(2a_1 + (n-1) \cdot d\right)
\][/tex]
- Here, [tex]\(n = 69\)[/tex].
5. Calculations:
- Without specific information on [tex]\(a_1\)[/tex] or [tex]\(d\)[/tex], assume [tex]\(a_1\)[/tex] (the "starting" term for calculation) is 69 for simplicity. Similarly assume [tex]\(d = 0\)[/tex] as the common difference doesn't impact without additional detail.
- Substitute in the known values:
[tex]\[
S_{69} = \frac{69}{2} \times \left(2 \cdot 69 + (69 - 1) \cdot 0\right)
\][/tex]
- Calculate [tex]\(S_{69}\)[/tex]:
[tex]\[
S_{69} = \frac{69}{2} \times 138 = 4761
\][/tex]
Thus, the sum of the first 69 terms of this arithmetic progression is 4761.
Here's a step-by-step explanation:
1. Understand the Terms and Definitions:
- The 35th term of the A-P is 69.
- We need to find the sum of the first 69 terms.
2. Use the Formula for the nth Term of an A-P:
- The formula for the nth term ([tex]\(a_n\)[/tex]) of an arithmetic progression is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.
3. Apply the Given Information:
- From the information, the 35th term ([tex]\(a_{35}\)[/tex]) is 69, so:
[tex]\[
a_{35} = a_1 + 34d = 69
\][/tex]
4. Find the Sum of the First 69 Terms:
- The formula for the sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) of an A-P is:
[tex]\[
S_n = \frac{n}{2} \left(2a_1 + (n-1) \cdot d\right)
\][/tex]
- Here, [tex]\(n = 69\)[/tex].
5. Calculations:
- Without specific information on [tex]\(a_1\)[/tex] or [tex]\(d\)[/tex], assume [tex]\(a_1\)[/tex] (the "starting" term for calculation) is 69 for simplicity. Similarly assume [tex]\(d = 0\)[/tex] as the common difference doesn't impact without additional detail.
- Substitute in the known values:
[tex]\[
S_{69} = \frac{69}{2} \times \left(2 \cdot 69 + (69 - 1) \cdot 0\right)
\][/tex]
- Calculate [tex]\(S_{69}\)[/tex]:
[tex]\[
S_{69} = \frac{69}{2} \times 138 = 4761
\][/tex]
Thus, the sum of the first 69 terms of this arithmetic progression is 4761.
