List of top Mathematics Questions on Sum of First n Terms of an AP

For positive integers \( n \), if \( 4 a_n = \frac{n^2 + 5n + 6}{4} \) and $$ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} $$ 

Let \[ f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{\tan \left( \frac{x}{2^{r+1}} \right) + \tan^3 \left( \frac{x}{2^{r+1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{r+1}} \right)} \right) \] Then, \( \lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)} \) is equal to:

$$ \lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \ldots + ((n-1)^2 - (n-1))}{(1^3 + 2^3 + \ldots + n^3) - (1^2 + 2^2 + \ldots + n^2)} $$ is equal to: 

If \(S_n=3+7+11....\) upto \(n\) terms and \(40<\frac {6}{n(n+1)}\displaystyle\sum_{k=1}^n S_k<45\). Then \(n\) is

Let the first term of a series be \( T_1 = 6 \) and its \( r^\text{th} \) term \( T_r = 3T_{r-1} + 6^r \), \( r = 2, 3, \dots, n \). If the sum of the first \( n \) terms of this series is \[ \frac{1}{5} \left(n^2 - 12n + 39\right) \left(4 \cdot 6^n - 5 \cdot 3^n + 1\right), \] then \( n \) is equal to ______.

Let \(a_1,a_2,a_3\), ..., an, be in A. P. and \(S_n\) denotes the sum of first \(n\) terms of this A. P. is \(S_{10}\) =\(390, \frac{a_{10}}{a_{50}} =\frac{15}{7}\), then \(S_{15} -S_5 =\) _________.

Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic progression. If \( S_{20} = 790 \) and \( S_{10} = 145 \), then \( S_{15} - S_5 \) is:

The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ , is_________

The sum $1+ \frac{1^{3} +2^{3}}{1+2} + \frac{1^{3}+2^{3}+3^{3}}{1+2+3} +.... + \frac{1^{3} +2^{3}+3^{3} +....+15^{3}}{1+2+3+...+15} - \frac{1}{2} \left(1+2+3+...+15\right)$

In AP, \( a_k=5k+1 \). Find sum of first 100 terms

The sum of odd integers from 1 to 2001 is

In AP, \( a_k=5k+1 \). Find sum of first 100 terms

The sum of odd integers from 1 to 2001 is

The sum of odd integers from 1 to 2001 is:

If $a_1, a_2, a_3, a_4$ are in A.P., then $\frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \frac{1}{\sqrt{a_3} + \sqrt{a_4}} =$

The sum $\displaystyle\sum^{10}_{r=1}(r^2 + 1) \times (r!)$ is equal to :

Find the sum of the following APs:

1. \(2, 7, 12, .......,\) to \(10\) terms.
2. \(–37, –33, –29,.......,\) to \(12\) terms.
3. \(0.6, 1.7, 2.8, ......., \)to \(100\) terms.
4. \(\frac {1}{15}, \frac {1}{12}, \frac {1}{10}, .......,\) to \(11\) terms.

Find the sums given below :

1. \(7 + 10\frac 12+ 14 + ....... + 84\)
2. \(34 + 32 + 30 + ....... + 10\)
3. \(–5 + (–8) + (–11) + ....... + (–230)\)

A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of \(\frac 14\) m and a tread of \(\frac 12\) m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace.

The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are \(2\frac 12\) m apart, what is the length of the wood required for the rungs?

The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

In an AP:

1. given a = 5, d = 3, an = 50, find n and S_n
2. given a = 7, a₁₃ = 35, find d and S₁₃.
3. given a₁₂ = 37, d = 3, find a and S₁₂.
4. given a₃ = 15, S₁₀ = 125, find d and a₁₀.
5. given d = 5, S₉ = 75, find a and a₉.
6. given a = 2, d = 8, S_n = 90, find n and a_n .
7. given a = 8, a_n = 62, S_n = 210, find n and d.
8. given a_n = 4, d = 2, S_n = –14, find n and a.
9. given a = 3, n = 8, S = 192, find d.
10. given l = 28, S = 144, and there are total 9 terms. Find a.

In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line (see Fig. 5.6).

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?