Find the largest real value of `x` such that `sum_(k=0)^4((3^(4-k))/((4-k)!))((x^k)/(k !))=(32)/3`.

Answer

1

Explanation

`underset(k=0)overset(4)sum((3^(4-k))/((4-k)!))((x^(k))/(k!))`
`=underset(k=0)overset(4)sum((4!)/((4-k)!k!)3^(4-k).x^(4).(1)/(4!))`
`= underset(k=0)overset(4)sum(.^(4)C_(k).3^(4-k).x^(4))/(04!)`
`= ((3+x)^(4))/(4!)`
According to the question ,
`((3+x)^(4))/(4!) = 32/3`
or `(3+x)^(4) = 256`
or `x + 3 = 256`
or `x = 1`