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Kay [80]
2 years ago
11

I) Find the first 4 terms in the expansion of

Mathematics
1 answer:
Katyanochek1 [597]2 years ago
6 0

Answer:

(i)  64+192 x^2+240 x^4+160x^6

(ii) 1072

Step-by-step explanation:

<u>Part (i)</u>

Using Binomial series formula:

(2+x^2)^6=2^6+6C1 \cdot 2^{6-1}\cdot x^2+6C2 \cdot 2^{6-2}\cdot(x^2)^2+6C3 \cdot 2^{6-3}\cdot(x^2)^3

              =64+6 \cdot 32\cdot x^2+15 \cdot 16\cdot x^4+20 \cdot 8 \cdot x^6

              =64+192 x^2+240 x^4+160x^6

<u>Part (ii)</u>

<u />(1-\frac{3}{x^2})^2=(1-\frac{3}{x^2})(1-\frac{3}{x^2})

               =1-\dfrac{6}{x^2}+\dfrac{9}{x^4}

(2+x^2)^6(1-\frac{3}{x^2})^2=(64+192 x^2+240 x^4+160x^6)\left(1-\dfrac{6}{x^2}+\dfrac{9}{x^4} \right)

=64-\dfrac{384}{x^2}+\dfrac{576}{x^4}+192x^2-1152+\dfrac{1728}{x^2}+240x^4-1440x^2+2160+...

There is no need to keep expanding since the remaining will include the variable.

Therefore, the term independent of x = 64 - 1152 + 2160 = 1072

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