The sum of the coefficients of the first three terms in the expansion of(x−x23)m,x=0,m being a natural number, is 559. Find the term of the expansion containing x3.

Solution: representing as sum of first three terms .mC0(−3)0+.mC1(−3)1+.mC2(−3)2=559 1+m(−3)+2m(m−1)⋅9=559 −3m+29m2−9m=558 32m2−52m=186 3m2−5m−372=0 m=65±25+12⋅372 =65+4464+25 =65+67672=12 to find the term for x3 so,m−3r=3 r=3 so fourth term will be takent4=.12C3(−3)x3 =12⋅11⋅3⋅210⋅(−27x3) =−5940x3

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The sum of the coefficients of the first three terms in the ex

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The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x ^{2}})^{m}, x \neq = 0,$ m being a natural number, is 559. Find the term of the expansion containing $x^{3}$ .

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Scarlett Discussed

The sum of the coefficients of the first three terms in the expansion of

(x - \frac{3}{x ^{2}})^{m}, x \neq = 0,

m being a natural number, is 559. Find the term of the expansion containing

x^{3}

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Text solutionVerified

Solution: representing as sum of first three terms

.^{m} C_{0} (- 3)^{0} + .^{m} C_{1} (- 3)^{1} + .^{m} C_{2} (- 3)^{2} = 559

1 + m (- 3) + \frac{m ( m - 1 )}{2} \cdot 9 = 559

- 3 m + \frac{9 m ^{2} - 9 m}{2} = 558

3 \frac{m ^{2}}{2} - 5 \frac{m}{2} = 186

3 m^{2} - 5 m - 372 = 0

m = \frac{5 \pm 25 + 12 \cdot 372}{6}

= \frac{5 + 4464 + 25}{6}

= \frac{5 + 67}{6} \frac{72}{6} = 12

to find the term for

x^{3}

so,

m - 3 r = 3

r = 3

so fourth term will be taken

t_{4} = .^{12} C_{3} (- 3) x^{3}

= 12 \cdot 11 \cdot \frac{10}{3 \cdot 2} \cdot (- 27 x^{3})

= - 5940 x^{3}

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Question Text	The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x ^{2}})^{m}, x \neq = 0,$ m being a natural number, is 559. Find the term of the expansion containing $x^{3}$ .
Answer Type	Text solution:1
Upvotes	150

The sum of the coefficients of the first three terms in the expansion of(x−x23​)m,x=0,m being a natural number, is 559. Find the term of the expansion containing x3.

Text solutionVerified

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The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x ^{2}})^{m}, x \neq = 0,$ m being a natural number, is 559. Find the term of the expansion containing $x^{3}$ .