Question

(1)(a) Use the generalized binomial expansion to expand (1 + x)ā up to the x® and hence

determine (1.05). to 5 decimal places
10 marks​

Answer 1

The question is incomplete! Complete question along with answer and step by step explanation is provided below.

Question:

Use the generalized binomial expansion to expand (1 + x)^a up to the a = 5 and hence determine (1.05)^5. to 5 decimal places.

Answer:

Using the binomial theorem

`(1 .05)^5 = 1.27628`

Explanation:

The binomial theorem is given by

`(a +b)^n = \binom{n}{0} a^nb^0 + \binom{n}{1} a^(n-1)b^1....+ \binom{n}{n} a^0b^n`

For the given case, we have

`a = 1 \\\\b = 0.05 \\\\n = 5 \\\\`

So,

`(1 +0.05)^5 = \binom{5}{0} (1)^5(0.05)^0 + \binom{5}{1} (1)^4(0.05)^1 + \binom{5}{2} (1)^3(0.05)^2 + \binom{5}{3} (1)^2(0.05)^3 + \binom{5}{4} (1)^1(0.05)^4 + \binom{5}{5} (1)^0(0.05)^5`

`(1 +0.05)^5 = (1)(1)^5(0.05)^0 + (5) (1)^4(0.05)^1 + (10) (1)^3(0.05)^2 + (10) (1)^2(0.05)^3 + (5) (1)^1(0.05)^4 + (1) (1)^0(0.05)^5`

`(1 +0.05)^5 = 1 + 0.25 + 0.025 + 0.00125 + 0.00003125 + 0.0000003125`

`(1 + 0.05)^5 = 1.27628`

Therefore, using the binomial theorem

`(1 .05)^5 = 1.27628`