Question

$\begin{array}{c}\text{Let}g\left(x\right)=x^6+a{x}^5+b{x}^4+c^3+d{x}^2+ex+f\text{be a polynomial such that}\\ g\left(1\right)=1,g\left(2\right)=2,g\left(3\right)=3,g\left(4\right)=4,g\left(5\right)=5\text{and}g\left(6\right)=6,\text{then the value of}g\left(7\right)\text{is}\end{array}$

• A. $zero$
• B. $7$
• C. $10$
• D. $727$

Answer 1

The correct option is C $10$

$g\left(x\right)=x^6+a{x}^5+b{x}^4+c^3+d{x}^2+ex+f$

$g\left(1\right)=1,g\left(2\right)=2,g\left(3\right)=3,g\left(4\right)=4,g\left(5\right)=5,g\left(6\right)=6,g\left(7\right)=7$

Let $f\left(0\right)=3$

${\sum }_{k=0}^7(-1{)}^k{(}_k^7)f(x-k)=0$

$f\left(7\right)=-{\sum }_{k=1}^7(-1{)}^k{(}_k^7)f(7-k)$

$=7f\left(6\right)-21f\left(5\right)+35f\left(4\right)-35f\left(3\right)+21f\left(2\right)-7f\left(1\right)+f\left(0\right)$

$f\left(7\right)=10$