In the expansion of $$\((a+b x)^{7}\)$$, where $$\(a\)$$ and $$\(b\)$$ are non-zero constants, the coefficients of $$\(x, x^{2}\)$$ and $$\(x^{4}\)$$ are the first, second and third terms respectively of a geometric progression. Find the value of $$\(\frac{a}{b}\)$$ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_w20_qp_13 Year：2020 Question No：5
Answer:
\(\left[7 C 1 a^{6} b(x)\right], \quad\left[7 C 2 a^{5} b^{2}\left(x^{2}\right)\right], \quad\left[7 C 4 a^{3} b^{4}\left(x^{4}\right)\right]\)
\(\frac{7 C 2 a^{5} b^{2}\left(x^{2}\right)}{7 C 1 a^{6} b(x)}=\frac{7 C 4 a^{3} b^{4}\left(x^{4}\right)}{7 C 2 a^{5} b^{2}\left(x^{2}\right)} \rightarrow \frac{21 a^{5} b^{2}}{7 a^{6} b}=\frac{35 a^{3} b^{4}}{21 a^{5} b^{2}}\)
\(\frac{a}{b}=\frac{5}{9}\)
\(\frac{7 C 2 a^{5} b^{2}\left(x^{2}\right)}{7 C 1 a^{6} b(x)}=\frac{7 C 4 a^{3} b^{4}\left(x^{4}\right)}{7 C 2 a^{5} b^{2}\left(x^{2}\right)} \rightarrow \frac{21 a^{5} b^{2}}{7 a^{6} b}=\frac{35 a^{3} b^{4}}{21 a^{5} b^{2}}\)
\(\frac{a}{b}=\frac{5}{9}\)
Knowledge points:
1.6.1 use the expansion of , where is a positive integer (Including the notations and n!) (Knowledge of the greatest term and properties of the coefficients are not required.)
1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)
Solution:
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