Using binomial theorem, prove that 3^ 2n+2 -8n-9 is divisible by … - Vidyadip

Question

Using binomial theorem, prove that $3^{2\text{n}+2}-8\text{n}-9$ is divisible by 64 where $\text{n}\in\text{N}.$

✓

Answer

$3^{2\text{n}+2}-8\text{n}-9$
$=3^{2(\text{n}+1)}-8\text{n}-9$
$=9^{\text{n}+1}-8\text{n}-9$
$=(1+8)^{\text{n}+1}-8\text{n}-9$
$\Big({^{\text{n}+1}}\text{C}_0+{^{\text{n}+1}}\text{C}_18^1+{^{\text{n}+1}}\text{C}_28^2+.....+{^{\text{n}+1}}\text{C}_{\text{n}+1}8^{\text{n}+1}\Big)-8\text{n}-9$
$\Big(1+8(\text{n}+1)64^{\text{n}+1}\text{C}_2+64(8)^{\text{n}-1}\Big)-8\text{n}-9$
$=64\Big({^{\text{n}+1}\text{C}}_2+......+8^{\text{n}-1}\Big)$
Thus, $3^{2\text{n}+2}-8\text{n}-9$ is divisible by 64.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Binomial Theorem→Binomial Theorem — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Sketch the graphs of the following functions:
$\text{f(x)}=2\sec\pi\text{x}$

calculate the mean deviation from the mean for the following data:
$57, 64, 43, 67, 49, 59, 44, 47, 61, 59$

$\text{a}(\cos\text{C}-\cos\text{B})=2(\text{b}-\text{c})\cos^2\frac{\text{A}}{2}.$

If $\text{f(x)}=\frac{1}{1-\text{x}},$ show that $\text{f}\big[\text{f}\{\text{f(x)}\}\big]=\text{x}$

The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observation were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observation were omitted.

Find the equation of a straight line passing through the point of intersection of 2x + 3y + 1 = 0 and 3x - 5y - 5 = 0 and equally inclined to the axes.

In what ratio is the line joining the points (2, 3) and (4, -5) divided by the line passing through the points (6, 8) and (-3, -2).

If $\Big(\frac{1+\text{i}}{1-\text{i}}\Big)^3-\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^3=\text{x}+\text{iy,}$ find (x, y)

Reduce each of the following expressions to the sine and cosin of a single expression:
$\sqrt{3}\sin\text{x}-\cos\text{x}$

Each set X, contains 5 elements and each set Y, contains 2 elements and $\bigcup^\limits{20}_{\text{r=1}}\text{X}_\text{r}=\text{S =}\bigcup\limits^\text{n}_\text{r=1}\text{Y}_\text{r}.$ If each element of S belongs to exactly 10 of the $\text{X}'^\text{s}_\text{r}$ and to exactly 4 of $\text{Y}'^\text{s}_\text{r}$, then find the value of n.