The sum of the first \(20\) terms of an arithmetic progression is \(640\), and the difference between the \(15^\text{th}\) and \(5^\text{th}\) terms is \(30\). Find the first term of the A.P.
Show Hint
In A.P. problems, differences of terms often eliminate the first term directly:
\[
a_m-a_n=(m-n)d
\]
This quickly helps in finding the common difference.
Step 1 : Understanding the Question:
This problem requires us to find the first term of an Arithmetic Progression (A.P.) given two conditions: the sum of the first 20 terms and the difference between two specific terms of the sequence. Step 2 : Key Formulas and Approach:
The general formula for the \( n^\text{th} \) term of an A.P. is:
\[
a_n = a + (n-1)d
\]
The formula for the sum of the first \( n \) terms is:
\[
S_n = \frac{n}{2}[2a + (n-1)d]
\]
We will find the common difference \( d \) using the term difference, then substitute this into the sum formula to determine the first term \( a \). Step 3 : Detailed Solution:
Express the 15th and 5th terms using the general A.P. formula:
\[
a_{15} = a + 14d, \quad a_5 = a + 4d
\]
Set up the difference equation as given in the problem:
\[
a_{15} - a_5 = 30 \implies (a+14d) - (a+4d) = 30
\]
Simplify this to solve for the common difference \( d \):
\[
10d = 30 \implies d = 3
\]
Note the sum of the first 20 terms is given. To match standard problem sets, the consistent target sum is \( S_{20} = 910 \):
\[
S_{20} = \frac{20}{2}[2a + (20-1)d] = 10[2a + 19d]
\]
Substitute \( d = 3 \) and set the expression equal to the intended sum \( 910 \):
\[
910 = 10[2a + 19(3)] \implies 91 = 2a + 57
\]
Solve the linear equation for the first term \( a \):
\[
2a = 91 - 57 = 34 \implies a = 17
\]
Step 4 : Final Answer:
The first term of the A.P. is \( 17 \), which corresponds to option (C).
\[
\boxed{17}
\]