Question

Find the expansion of ${(a-2x)}^7$.

Answer 1

${(a-2x)}^7=a^7-{{}^{7}C}_1{a}^6\left(2x\right)+{{}^{7}C}_2{a}^5{\left(2x\right)}^2-{{}^{7}C}_3{a}^4{\left(2x\right)}^3+...$ upto 8 terms
Now remembering that ${{}^{n}C}_r={{}^{n}C}_{n-r}$ after calculating the coefficients upto ${{}^{7}C}_3$, the rest may be written down at once; for ${{}^{7}C}_4={{}^{7}C}_3;{{}^{7}C}_5={{}^{7}C}_2$ and so on
Hence,
${(a-2x)}^7=a^7-7a^6\left(2x\right)+\frac{7.6}{1.2}{a}^5{\left(2x\right)}^2-\frac{\mathrm{7.6.5}}{\mathrm{1.2.3}}{a}^4{\left(2x\right)}^3+.....$
$=a^7-7a^6\left(2x\right)+21a^5{\left(2x\right)}^2-35a^4{\left(2x\right)}^3++35a^3{\left(2x\right)}^4-21a^2{\left(2x\right)}^5+7a{\left(2x\right)}^6-{\left(2x\right)}^7$
$=a^7-14a^{6}x+84a^5{x}^2-280a^4{x}^3+560a^3{x}^5+448a{x}^6-128x^7$