What is {1}{{r = 0}{∑}} ⁿ+r}Cₙ equal to?

Correct Answer is : n+2C1ⁿ + 2}C₁n+2C1

Correct Answer is : n+2C1ⁿ + 2}C₁n+2C1

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UPSC NDA I 2015 Mathematics Paper

Question : 19 of 120

Marks: +1, -0

What is

\overset{1}{\underset{r = 0}{∑}} ^{n+r}C_{n}

equal to?

Solution:

Concept:

The sum is evaluated by expanding the terms using the identity

^{n}C_{n} = 1

and

^{n+1}C_{n} = n+1

Explanation:

The given sum is

\overset{1}{\underset{r = 0}{\sum}} \, {}^{n+r}C_{n}

Write the sum for each value of

r

r = 0

gives

^{n+0}C_{n} = 1

, and

r = 1

gives

^{n+1}C_{n} = n+1

Add these two terms:

1 + (n+1) = n+2

Now

n+2

can be written as

^{n+2}C_{1}

because

^{n+2}C_{1} = n+2

Thus the sum equals

^{n+2}C_{1}

Answer:

^{n+2}C_{1}

(Option A).

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