1.

Using Binomial theorem find the values of following(i) (99)5(ii) (1.1)6

Answer»

(i) (99)5

99 = (100- 1)

995 = (100- 1)5 = [100+ (- 1)]5

By using Binomial theorem

995 = 5C0 (100)5 (-1)0 + 5C, (100)4 (-1)1
+ 5C2 (100)3 (- 1)2 + 5C3 (100)2 (- 1)3
+ 5C4 (100) (-1)4 + 5C5 (100)0 (-1)5

= 1 x 10000000000 x 1 + 5 x 100000000
× (- 1) + 10 x 1000000 × 1 + 10 × 10000
× (- 1)+ 5 x 100 × 1 + 1 × 1 × (-1)

= 10000000000 – 500000000 +
10000000 – 100000 + 500 – 1

= 10010000500 – 500100001

= 9509900499

Hence, (99)5 = 9509900499

(ii) (1.1)6

(1.1)6 = (1 +0.1)6

= 6C0(1)6 (0.1)0 + 6C1 (1)5 (0.1)1
+ 6C2 (1)4 (0.1)2 + 6C3 (1)3 (0.3)3
+ 6C4 (1)2 (0.1)4 + 6C5 (1)1 (0.1)5 + 6C6 (1)0 (0.1)6

= (1 x 1 x 1) + (6 x 1 x 0-1) + {15 x 1 x (0.1)2}
+ {20 x 1 x (0.1)3} + {15 x 1 x (0.1)4}
+ {6 x 1 x (0.1)5} + {1 x 1 x (0.1)6}

= 1 + 6 x 0.1 + 15 x (0.1)2 + 20 x (0.1)3
+ 15 x (0.1)4+ 6 x (0.1)5 + (0.1)6

= 1 + 0.6 + 15 x 0.01 + 20 x 0.001 + 15 x 0.0001
+ 6 x 0.00001 +0.000001

= 1+0.6 + 0.15 + 0020 + 0-0015 + 0.00006 + 0000001

= 1.771561


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