Question

If the sum of the first 8 terms of an A.P. is 64 and the sum of its first 17 terms is 289, then find the first term and the common difference of the progression.

Hint:

To find unknown terms in an A.P., use the sum formula \( S_n = \frac{n}{2}[2a + (n - 1)d] \) for different values of \( n \) to form simultaneous equations.

Answer 1

Step 1: Recall the formula for the sum of n terms of an A.P.
\[ S_n = \frac{n}{2} [2a + (n - 1)d] \]
Step 2: Use the given information.
For \( n = 8 \): \[ S_8 = 64 \Rightarrow 64 = \frac{8}{2} [2a + 7d] \Rightarrow 4(2a + 7d) = 64 \] \[ 2a + 7d = 16 \quad \text{(i)} \] For \( n = 17 \): \[ S_{17} = 289 \Rightarrow 289 = \frac{17}{2} [2a + 16d] \] \[ 34a + 272d = 578 \Rightarrow 2a + 16d = 34 \quad \text{(ii)} \]
Step 3: Solve equations (i) and (ii).
From (i): \( 2a + 7d = 16 \) From (ii): \( 2a + 16d = 34 \) Subtract (i) from (ii): \[ (2a + 16d) - (2a + 7d) = 34 - 16 \] \[ 9d = 18 \Rightarrow d = 2 \]
Step 4: Substitute \( d = 2 \) in (i).
\[ 2a + 7(2) = 16 \Rightarrow 2a + 14 = 16 \Rightarrow 2a = 2 \Rightarrow a = 1 \] Step 5: Conclusion.
The first term \( a = 1 \) and the common difference \( d = 2 \). \[ \boxed{a = 1, \; d = 2} \]