Question

Find the sum $(x+2{)}^{n-1}+(x+2{)}^{n-2}(x+1)+(x+2{)}^{n-3}(x+1{)}^2+\cdots +(x+1{)}^{n-1}$

• A. $(x+2{)}^{n-2}-(x+1{)}^n$
• B. $(x+2{)}^{n-1}-(x+1{)}^{n-1}$
• C. $(x+2{)}^n-(x+1{)}^n$
• D. none of these

Answer 1

The correct option is C $(x+2{)}^n-(x+1{)}^n$

Let,

$S=(x+2{)}^{n-1}+(x+2{)}^{n-2}(x+1)+(x+2{)}^{n-3}(x+1{)}^2+\cdots +(x+1{)}^{n-1}$

We know that,

$\frac{(a{)}^n-(b{)}^n}{\left(a\right)-\left(b\right)}=a^{n-1}+a^{n-2}b+a^{n-3}{b}^2+\cdots +b^{n-1}$

$\therefore \frac{(x+2{)}^n-(x+1{)}^n}{(x+2)-(x+1)}=(x+2{)}^{n-1}+(x+2{)}^{n-2}(x+1)+(x+2{)}^{n-3}(x+1{)}^2+\cdots +(x+1{)}^{n-1}\Rightarrow S=\frac{(x+2{)}^n-(x+1{)}^n}{(x+2)-(x+1)}\Rightarrow S=(x+2{)}^n-(x+1{)}^n$