The value of ^n_ r=0 a_ 2r-1 is: - Vidyadip

MCQ

The value of $\sum\limits^\text{n}_{\text{r}=0}\text{a}_{2\text{r}-1}$ is:

A
$9^n- 1$
✓
$9^n + 1$
C
$9^n - 2$
D
$9^n+ 2$

✓

Answer

Correct option: B.

$9^n + 1$

$\left(1+4 x+4 x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots\left(a_{2 n} x^{2 n}\right)$
Substituting $x=1$ we get
$9^n=a_0+a_1+a_2+\ldots\left(a_{2 n}\right)$
Substituting $x=-1$ we get
$1=a_0-a_1+a_2-a 3 \ldots\left(a_{2 n}\right)$
Adding both we get
$2\left(a_0+a_2+a_4+\ldots a^{2 n}\right)=9^n+1$
Hence
$\sum^{\text{n}}_{\text{k}=0}\text{a}_{2\text{k}}=9^{\text{n}}+1$

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