Question

Let \( a_1, a_2, a_3, \dots \) be in an arithmetic progression of positive terms.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).  
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.

Answer 1

Correct Answer: 910

Given:

\[ d \rightarrow \text{common difference.} \]

The general term:

\[ A_k = kd \left[ 2a + (2k - 1)d \right] \]

Given:

\[ A_3 = -153 \] \[ \Rightarrow 153 = 13d \left[ 2a + 5d \right] \]

Simplifying:

\[ 51 = d \left[ 2a + 5d \right] \quad \dots (1) \]

Also, given:

\[ A_5 = -435 \] \[ 435 = 5d \left[ 2a + 9d \right] \]

Simplifying:

\[ 87 = d \left[ 2a + 9d \right] \quad \dots (2) \]

Subtracting equation (1) from equation (2):

\[ 36 = 4d^2 \] \[ d = 3, \quad a = 1 \]

Finally:

\[ a_{17} - A_7 = 49 - \left[ -7.3 \left[ 2 + 39 \right] \right] = 910 \]