Sequences & Series
Telescoping or product series
nta_pyq_2025_apr
Grade 12
Question:
4 1 1 1$\pi If + + +$$\ldots..$$\infty =, 4 4 4 1 2 3 90 1 1 1 + + +$$\ldots..$$\infty =$$\alpha, 4 4 4 1 3 5 1 1 1 + + +$$\ldots.$$\infty =$$\beta, 4 4 4 2 4 6 then$$\alpha$$\beta is equal to$
Step-by-Step Solution
Key Concept: Use the visible$product/series$structure to transform the expression into a telescoping or standard finite sum.
If 1$4 + 1$$4 + 1$4 +$\ldots..$$\infty =$$\pi 90.... (i) 1 2 3$(3)$1 1 1$$\beta = + + +$$\ldots, 4 4 4 2 4 6 1 1 1 1 =$[ + + +$\ldots..$], 4 4 4 16 1 2 3 4 1$\pi =$$\times$ 16 90 using (ii) ________...(ii) 1 1 1$\alpha = + + +$$\ldots..$$\infty 4 4 4 1 3 5 1 1 1 1 1 ( + + + + +$$\ldots..) 4 4 4 4 4 1 2 3 4 5 1 1$1 - ($+ + +$$\ldots..) 4 4 4 2 4 6 4 4 using (i) and (ii)$]$\pi 1$$\pi$$\alpha = -$$\times$ [ 90 16 90 4$16 - 1$15$\pi 4 4$$\alpha =$$\times$$\pi =$$\$pi = 1$6$\times 90 16 \times 90 96 4$\pi$$\alpha 96 16$\times 90 ∴ = = = 15 4$\beta$$\pi 96 16$\times 90
Correct Answer: 3