Question

if Kth trm iss he greatest term in the square an=n/70+n2 then k is

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (B)

8

Answer 2

We are given the sequence

$$a_n = \frac{n}{70 + n^2}$$

Step 1: Treat n as a continuous variable and differentiate.

$$f(n) = \frac{n}{70+n^2}$$

Differentiate using the quotient rule:

$$f'(n) = \frac{(70+n^2)(1) - n(2n)}{(70+n^2)^2} = \frac{70+n^2-2n^2}{(70+n^2)^2} = \frac{70 - n^2}{(70+n^2)^2}$$

Set $f'(n) = 0$:

$$70 - n^2 = 0 \quad \Rightarrow \quad n^2 = 70 \quad \Rightarrow \quad n = \sqrt{70} \approx 8.37$$

Step 2: Determine the integer maximum.

Since $n$ is an index, check the nearby integers:

• For $n = 8$:

  $$a_8 = \frac{8}{70 + 64} = \frac{8}{134} \approx 0.05970$$

• For $n = 9$:

  $$a_9 = \frac{9}{70 + 81} = \frac{9}{151} \approx 0.05960$$

Thus, $a_8 > a_9$.

Conclusion:

The greatest term occurs at $n = 8$. Therefore, the $k^\text{th}$ term which is the greatest corresponds to $k = 8$.