Sum the series: 1 + 2.2 + 3.22 + ..... + 100.299(a) 99.2100 (b) 100.2

1.	Sum the series: 1 + 2.2 + 3.22 + ..... + 100.299(a) 99.2100 (b) 100.2100(c) 99.2100 + 1 (d) 1000.2100
Answer» (c) 99.2¹⁰⁰+1 1 + 2.2 + 3.2² + 4.2³ + ..... + 100.2⁹⁹ is clearly an AGP with A.P. : 1 + 2 + 3 + ..... 100 and G.P. : 1 + 2 + 2² + ..... + 2⁹⁹. Let S = 1 + 2.2 + 3.2² + 4.2³ + ..... + 100.2⁹⁹ ∴ 2S = 2 + 2.2² + 3.2³ + ..... + 99.2⁹⁹ + 100.2¹⁰⁰ ∴ S – 2S = (1 + 2 + 2² + 2³ + ..... 2⁹⁹) – 100.2¹⁰⁰ ⇒ -S = \(\frac{(2^{100}-1)}{2-1}\) - 100.2¹⁰⁰ (∵ S_n = \(\frac{a(r^n-1)}{r-1}\) and this is a G.P. with 100 terms, a = 1, r = 2) = 2¹⁰⁰ – 1 – 100.2¹⁰⁰ S = 2¹⁰⁰ . 100 – 2¹⁰⁰ + 1 = 2¹⁰⁰ . (100 – 1) + 1 = 99.2¹⁰⁰ + 1.

Sum the series: 1 + 2.2 + 3.22 + ..... + 100.299(a) 99.2100 (b) 100.2100(c) 99.2100 + 1 (d) 1000.2100

Answer»

(c) 99.2¹⁰⁰+1

1 + 2.2 + 3.2² + 4.2³ + ..... + 100.2⁹⁹ is

clearly an AGP with A.P. : 1 + 2 + 3 + ..... 100

and G.P. : 1 + 2 + 2² + ..... + 2⁹⁹.

Let S = 1 + 2.2 + 3.2² + 4.2³ + ..... + 100.2⁹⁹

∴ 2S = 2 + 2.2² + 3.2³ + ..... + 99.2⁹⁹ + 100.2¹⁰⁰

∴ S – 2S = (1 + 2 + 2² + 2³ + ..... 2⁹⁹) – 100.2¹⁰⁰

⇒ -S = \(\frac{(2^{100}-1)}{2-1}\) - 100.2¹⁰⁰

(∵ S_n = \(\frac{a(r^n-1)}{r-1}\) and this is a G.P. with 100 terms, a = 1, r = 2)

= 2¹⁰⁰ – 1 – 100.2¹⁰⁰

S = 2¹⁰⁰ . 100 – 2¹⁰⁰ + 1 = 2¹⁰⁰ . (100 – 1) + 1

= 99.2¹⁰⁰ + 1.