Let a1, a2, a3, ….. be in arithmetic progression of positive terms. Let Ak = a21 – a22 + a23 – a24 +…..+ a22k–1 – a22k If A3 = – 153, A5 = – 435 and a21 + a22 + a23 = 66, then a17

Engineering

Mathematics

Arithmetic Progression and Its Properties

Question

Let a₁, a₂, a₃, ….. be in arithmetic progression of positive terms. Let A_k = a²₁ – a²₂ + a²₃ – a²₄ +…..+ a²_2k–1 – a²_2kIf A₃ = – 153, A₅ = – 435 and a²₁ + a²₂ + a²₃ = 66, then a₁₇– A₇ is equal to_______.

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A_k = –d[a₁ + a₂ + a₃ +,………+ a_2k]
A₃ = – 153

$\Rightarrow 2 a_{1} + 5 d = \frac{51}{d} . . . . . . . . . . (1)$

A₅ = – 435

$\Rightarrow 2 a_{1} + 9 d = \frac{87}{d} . . . . . . . . . . (2)$

From (1) & (2)
d = 3
Now, a₁
2 + (a₁ + d)² + (a₁ + 2d)² = 66
a₁² + (a₁ + 3)² + (a₁ + 6)² = 66
a₁² + 6a₁ – 7 = 0
(a₁ + 7) (a₁ – 1) = 0
⇒ a₁ = 1
a₁₇ = a₁ + 16d = 1 + 16 × 3 = 49
A₇ = – 3[7(2 × 1 + 13 × 3)] = – 861
a₁₇ – A₇ = 49 + 861 = 910