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Question

Question: If S<sub>1</sub> = {1}, S<sub>2</sub> = {3, 5}, S<sub>3</sub> = {7, 9, 11}, S<sub>4</sub> = {13, 15,...

If S1 = {1}, S2 = {3, 5}, S3 = {7, 9, 11}, S4 = {13, 15, 17, 19} and so on, then the sum of all the numbers in S20 is

A

10000

B

10,100

C

8100

D

8000

Answer

8000

Explanation

Solution

First entry of each set form a series

1, 3, 7, 13, 21,………

Let tn be the nth term of the series

Sn = 1 + 3 + 7 + 13 + 21 + ………. + tn

Sn = 1 + 3 + 7 + 13 + ………. + tn–1 + tn

0 = 1 + 2 + 4 + 6 + 8 + …… + (tn – tn–1) – tn

tn = 1 + n12\frac{n–1}{2} [4 + (n – 2) .2]

= 1 + (n – 1). n = n2 – n + 1

tn = n2 – n + 1

\first term in S20 is = 400 – 20 + 1 = 381

Sum of the numbers in

S20 = 202\frac{20}{2} (762 + 19 × 2) = 8000