Question

Find the value of a and b if $1\times 5+2\times 5^2+3\times 5^3+.....x\times 5^x=\frac{(4x-1)5^{a+1}+b}{16}$

• A. a=x,b=2
• B. a=x, b=5
• C. a=0,b=1
• D. none of these

Answer 1

The correct option is B

a=x, b=5

Conventional Approach :
$S=1\times 5+2\times 5^2+3\times 5^3+....+x\times 5^x.....\left(i\right)$
$5S=1\times 5^2+2\times 5^3+.....+x\times 5^{(x+1}).........(ii)$
$(S-5S)=1\times 5+5^2(2-1)+5^3(3-2)+....+5^x(x-(x-1\left)\right)-x\times 5^{(x+1)}$
$-4S=5+5^2+5^3+....+5^x-x\times 5^{(x+1)}$
$4S=x\times 5^{(x+1)}-(5+5^2+5^3+.......+5^x)$
$S=\frac{x\times 5^{(x+1)}-(5+5^2+5^3+,,,+5^x)}{4}$
$\frac{(4x-1)5^{(a+1)}+b}{16}=\frac{x\times 5^{(x+1)}-(5+5^2+5^3+,,,+5^x)}{4}$
$\frac{(4x-1)5^{(a+1)}+b}{16}=\frac{x\times 5^{(x+1)}-\frac{\left(5^{(x+1)}\right)-5}{4}}{4}$
$\frac{(4x-1)5^{(a+1)}+b}{16}=\frac{(4x-1)5^{(x+1)}+5}{16}$,
Now we can compare both sides and we can determine a=x and b=5.
Go from answer options.
For answer option b.
When a=x and b=5. Take x=1 , then a=1 and b=5.
LHS =5.
RHS =$\frac{\left[3\right(25)+5]}{16}=5$