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

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:

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