Question

Let a­­₁ a₂, ….. be integers such that

a₁ − a₂ + a₃ − a₄ + ⋯ + (−1)^{(n - 1)}a_n = n, for all n ≥ 1

Then a₅₁ + a₅₂ + ….. + a₁₀₂₃ equals

1. -1
2. 1
3. 0
4. 10

Answer 1

Correct Answer - Option 2 : 1

Calculation:

a₁ − a₂ + a₃ − a₄ + ⋯ + (−1)^{(n - 1)}a_n = n

For n = 1	⇒ (−1)(1 - 1)a1 = 1 ⇒ a1 = 1	a1 = 1
For n = 2	⇒ a1 – a2 = 2 ⇒ 1 – a2 = 2 ⇒ a2 = –1	a2 = –1
For n = 3	⇒ a1 – a2 + a3 = 3 ⇒ 1 – (–1) + a3 = 3 ⇒ a3 = 1	a3 = 1
For n = 4	⇒ a1 − a2 + a3 − a4 = 4 ⇒ 1 – (–1) + 1 – a4 = 4 ⇒ a4 = –1	a4 = –1

We can say,

⇒ a_odd = 1

⇒ a_even = –1

For a₅₁ + a₅₂ + ….. + a₁₀₂₃, we have equal number of odd and even terms till a₁₀₂₂, so their sum will be 0.

⇒ Left term = a₁₀₂₃(odd term)

⇒ a₁₀₂₃ = 1 (as it is an odd term)

∴ a₅₁ + a₅₂ + ….. + a₁₀₂₃ equals 1.