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

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 ______.
\[\lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \ldots + ((n-1)^2 - (n-1))}{(1^3 + 2^3 + \ldots + n^3) - (1^2 + 2^2 + \ldots + n^2)}\]is equal to:
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:
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 \(a_1,a_2,a_3\), ..., 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 \(S_{15} -S_5 =\) _________.
The sum of first 10 terms of the arithmetic progression 34, 32, 30, .….. is
The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ , is_________
The sum $1+ \frac{1^{3} +2^{3}}{1+2} + \frac{1^{3}+2^{3}+3^{3}}{1+2+3} +.... + \frac{1^{3} +2^{3}+3^{3} +....+15^{3}}{1+2+3+...+15} - \frac{1}{2} \left(1+2+3+...+15\right)$
The sum $1+\frac{1+a}{2!} + \frac{1+a+a^{2}}{3!} + ... \infty $ is equal to
The sum value of the series $ \frac{3}{4} +\frac{5}{36} +\frac{7}{144} +\frac{9}{400} + ...\infty $ is
The sum $\displaystyle\sum^{10}_{r=1}(r^2 + 1) \times (r!)$ is equal to :
The value of the sum $1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5+\ldots$ upto $n$ terms is equal to
The sum of the series $\frac{1}{1\times2} ^{^{25}}C_{0} + \frac{1}{2\times3}^{^{25}}C_{1} + \frac{1}{3\times4} ^{^{25}}C_{2} + ......... + \frac{1}{26\times27} ^{^{25}}C_{25}$
The sum of the series $1+\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+.... $ upto $n $ terms is
The sum of the series $\log _{4} 2-\log _{8} 2+\log _{16} 2-\ldots$ is
The sum of the series $ \frac{1}{2.3}+\frac{1}{4.5}+\frac{1}{6.7}+....+\infty $ =
The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+......$ is
The sum of the series $1+\frac{1}{4\cdot2!}+\frac{1}{16\cdot 4!}+\frac{1}{64\cdot 6!} + \ldots \infty$ is :
For a positive integer n, let $f_n(\theta)=\Bigg(tan\frac{\theta}{2}\Bigg)(1+sec \theta)(1+sec 2\theta)(1+ sec 2^2 \theta)... (1+sec 2^n \theta), then$
Sum of the first n terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + ...$ is equal to
Find the sums given below :
  1. \(7 + 10\frac 12+ 14 + ....... + 84\)
  2. \(34 + 32 + 30 + ....... + 10\)
  3. \(–5 + (–8) + (–11) + ....... + (–230)\)
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?
A ladder has rungs 25 cm apart.
Fig. 5.7
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, .... as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take \(\pi = \frac {22}{7}\))
A spiral is made up of successive semicircles
Fig. 5.4

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R
Assertion (A) : (1) \(+(1+2+4)+(4+6+9) +(9+12+16)+\ldots+(81+90+100)=1000\) 

Reason (R) : \(\displaystyle \sum_{r=1}^{n}\left(r^{3}-[r-1]^{3}\right)=n^{3}\) for any natural number \(n\)

In the light of the above statements, choose the correct answer from the options given below

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.

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