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UPSC NDA 2 2022 Mathematics Solved Paper

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Question : 2 of 120

Marks: +1, -0

What is

\sum\limits_{r=0}^{n} 2^{r} C(n, r)

equal to ?

$2^{n}$
$3^{n}$
$2^{2n}$
$3^{2n}$

Solution:

Concept:

The sum

\sum_{r=0}^{n} 2^{r} C(n, r)

is the binomial expansion of

(1 + 2)^{n}

.

Explanation:

Recall the binomial theorem:

(x + y)^{n} = \sum_{r=0}^{n} C(n, r) x^{n-r} y^{r}

.

Substitute

x = 1

and

y = 2

in the theorem.

Then

(1 + 2)^{n} = \sum_{r=0}^{n} C(n, r) 1^{n-r} 2^{r}

.

Since

1^{n-r} = 1

, this simplifies to

\sum_{r=0}^{n} C(n, r) 2^{r}

.

Thus the given sum is exactly

(1 + 2)^{n}

which equals

3^{n}

.

Answer:

B.

3^{n}

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