Answer :
To solve the problem of finding the sum of the first 69 terms of an arithmetic progression (A-P), where we know that the 35th term is 69, we need additional information about the sequence such as the first term or the common difference.
### Understanding the Problem:
1. Given: The 35th term of the A-P is 69.
2. Find: The sum of the first 69 terms of the A-P.
### Step-by-Step Solution:
1. Arithmetic Progression Formula:
- The formula for the [tex]\( n \)[/tex]-th term of an arithmetic progression is:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
2. Using the Given Information:
- For the 35th term, [tex]\( a_{35} = a + 34d = 69 \)[/tex].
3. Sum of Terms Formula:
- The sum of the first [tex]\( n \)[/tex] terms [tex]\( (S_n) \)[/tex] of an arithmetic progression is given by:
[tex]\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\][/tex]
- We need the sum [tex]\( S_{69} \)[/tex], which requires the first term [tex]\( a \)[/tex] and the common difference [tex]\( d \)[/tex].
4. Conclusion:
- Without specific values for the first term [tex]\( a \)[/tex] or the common difference [tex]\( d \)[/tex], we cannot determine the exact sum of the first 69 terms.
- Additional data about either the first term or the common difference is necessary to compute the sum.
In summary, it's not possible to calculate the sum of the first 69 terms with the given information alone. We would need either the first term or the common difference to proceed further.
### Understanding the Problem:
1. Given: The 35th term of the A-P is 69.
2. Find: The sum of the first 69 terms of the A-P.
### Step-by-Step Solution:
1. Arithmetic Progression Formula:
- The formula for the [tex]\( n \)[/tex]-th term of an arithmetic progression is:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( d \)[/tex] is the common difference.
2. Using the Given Information:
- For the 35th term, [tex]\( a_{35} = a + 34d = 69 \)[/tex].
3. Sum of Terms Formula:
- The sum of the first [tex]\( n \)[/tex] terms [tex]\( (S_n) \)[/tex] of an arithmetic progression is given by:
[tex]\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\][/tex]
- We need the sum [tex]\( S_{69} \)[/tex], which requires the first term [tex]\( a \)[/tex] and the common difference [tex]\( d \)[/tex].
4. Conclusion:
- Without specific values for the first term [tex]\( a \)[/tex] or the common difference [tex]\( d \)[/tex], we cannot determine the exact sum of the first 69 terms.
- Additional data about either the first term or the common difference is necessary to compute the sum.
In summary, it's not possible to calculate the sum of the first 69 terms with the given information alone. We would need either the first term or the common difference to proceed further.
