The Binomial Expansion

Edexcel IAL WMA12/01/P2 /June 2021/Q4 (The Binomial Expansion, Estimation)

• Find, in ascending powers of x, up to and including the term in x3, the binomial expansion of​​ 

2+x813

fully​​ simplifying each coefficient.​​ 

(4)​​ 

• Use the answer to part (a) to find an approximation for 2.012513​​ .Give your answer to 3 decimal places

(3)

Without calculating 2.012513​​ 

• state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.​​ 

(1)

SOLUTION​​ 

a- Using binomial theorem to find the first 4 terms of the expression.​​ 

Finding the first three terms separately and will then arrange them in the ascending order of power of x.​​ 

an=130213x80=1 x 213x 1=1892 

n1an-1 b=131212x81=13 x 212xx8=6656 x

n2an-2b2=132211x82=78 x 211xx264=2496 x2

n2an-3b3=133210x83=286 x210xx383=572 x3

2+x813=8192+6656 x+2496x2+572x3+…

b-​​ 

The question asks us to estimate the​​ value of​​ 2. 012513. To do so, we may equate the bases that is ​​ 2.0125 to​​ 2+x8.​​ 

2+x8=2. 0125

x8=0.0125

x=0.1

Now, let us substitute to the expansion we did in part a.​​ 

2+x813=8192 +6656 x+2496 x2+572 x3+… 

2+0.1813=8192+6656 0.1+2496 0.12+572 0.13

2. 012513 ≈8883.132

c- ​​ The estimate is​​ underestimate​​ as the expansion is shortened to just first 4 terms out of total 13 terms. Terms from x4​​ to x13​​ aren’t considered in the calculation above. And, since all the terms are positive, the answer would be greater of​​ 2. 012513​​ if further values of terms​​ x4​​ to x13​​ were considered.​​