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UPSC NDA II 2025 Mathematics Paper

Question Numbers: 23-24

For the following two (02) items:

Let

p=\overset{n}{\underset{j=1}{∑}}log_{10}2^{j}andq=\overset{n}{\underset{j=1}{∑}}log_{10}5^{j}

Question : 23 of 120

Marks: +1, -0

If p + q = 66 then which one of the following is correct?

Solution:

Concept:

The sum of logarithms with the same base can be simplified using the product rule. The sum of first n natural numbers is

\frac{n(n+1)}{2}

Explanation:

Given

p = \sum_{j=1}^{n} \log_{10} 2^{j}

and

q = \sum_{j=1}^{n} \log_{10} 5^{j}

Using

\log a^{j} = j \log a

, we get

p = \log_{10}2 \cdot \frac{n(n+1)}{2}

and

q = \log_{10}5 \cdot \frac{n(n+1)}{2}

Add them:

p+q = (\log_{10}2 + \log_{10}5) \cdot \frac{n(n+1)}{2} = \log_{10}(2 \cdot 5) \cdot \frac{n(n+1)}{2} = \frac{n(n+1)}{2}

Given

p+q = 66

, so

\frac{n(n+1)}{2} = 66

→

n(n+1) = 132

→

n^{2}+n-132=0

Solve the quadratic:

n = \frac{-1 \pm \sqrt{1+528}}{2} = \frac{-1 \pm 23}{2}

Thus

n = 11

(positive integer) or

n = -12

(invalid).

Since

n = 11

, it lies in the range

9 < n < 12

Answer:

Option C:

9 < n < 12

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