Shortcut Methods
JEE Mains :
1. Sum of n terms of an A.P.
$$S_n = \frac{n}{2}[2a_1 + (n - 1)d]$$
where (a_1) is the first term, (d) is the common difference, and (n) is the number of terms.
2. Sum of n terms of a G.P.
$$S_n = \frac{a_1(r^n - 1)}{r - 1}$$
where (a_1) is the first term, (r) is the common ratio, and (n) is the number of terms.
3. Sum of n terms of an arithmetic-geometric series.
$$S_n = \frac{a_1(r^n - 1)}{r - 1} - \frac{d(n - 1)}{2}$$
where (a_1) is the first term, (r) is the common ratio, (d) is the common difference, and (n) is the number of terms.
4. Sum of n terms of a harmonic series.
$$H_n = \sum\limits_{i=1}^n \frac{1}{i} \approx \ln n + \gamma$$
where (\gamma \approx 0.57721) is the Euler-Mascheroni constant.
5. Sum of n terms of a telescoping series.
Telescoping series are series of the form
$$\sum\limits_{i=1}^n (a_i - a_{i+1})$$
where (a_i) and (a_{i+1}) are consecutive terms of the series. The sum of a telescoping series can be found by simply subtracting the last term from the first term:
$$S_n = a_1 - a_{n+1}$$
6. Sum of n terms of a binomial series.
The binomial series for ((1+x)^n) is given by
$$(1+x)^n = \sum\limits_{k=0}^n \binom{n}{k}x^k$$
where (\binom{n}{k}) is the binomial coefficient. The sum of the first n terms of the binomial series is known as the binomial sum and can be calculated using the formula
$$S_n = \sum\limits_{k=0}^n \binom{n}{k}x^k = (1 + x)^n$$
7. Product of n terms of an A.P.
$$P_n = a_1 \cdot a_2 \cdot \ldots = \frac{a_1[a_1 + (n-1)d]^n}{a_1^n}$$
where (a_1) is the first term, (d) is the common difference, and (n) is the number of terms.
8. Product of n terms of a G.P.
$$P_n - a_1 \cdot a_2 \cdot a_3 \cdot \ldots \cdot a_n = a_1^n \cdot r^{n(n-1)/2}$$
where (a_1) is the first term, (r) is the common ratio, and (n) is the number of terms.
9. Product of n terms of an arithmetic-geometric series.
$$P_n = \frac{a_1(r^n - 1)}{r - 1} \cdot r^{\frac{n(n-1)}{2}}$$
where (a_1) is the first term, (r) is the common ratio, and (n) is the number of terms.
10. Product of n terms of a harmonic series.
There is no simple formula for the product of n terms of a harmonic series.
11. Find the sum of n terms of the sequence {a_n} defined by a_n = 3n - 1.
$$S_n = \sum\limits_{i=1}^n (3i-1) = 3\cdot \frac{n(n+1)}{2} - n$$
12. Find the sum of n terms of the sequence {a_n} defined by a_n = 1/n.
$$S_n = \sum\limits_{i=1}^n \frac{1}{i} \approx \ln n + \gamma$$
13. Find the sum of n terms of the sequence {a_n} defined by a_n = n^2 - 1.
$$S_n = \sum\limits_{i=1}^n (i^2 - 1) = \frac{n(n+1)(2n+2)}{6}$$
14. Find the sum of n terms of the sequence {a_n} defined by a_n = 2^n - 1.
$$S_n = \sum\limits_{i=1}^n (2^n - 1) = 2^n - 1$$
15. Find the sum of n terms of the sequence {a_n} defined by a_n = n!.
$$S_n = \sum\limits_{i=1}^n n! = (n+1)! - 1$$
CBSE Board Exams
1. Sum of n terms of an A.P.
$$S_n = \frac{n}{2}[2a_1 + (n - 1)d]$$
where (a_1) is the first term, (d) is the common difference, and (n) is the number of terms.
2. Sum of n terms of a G.P.
$$S_n = \frac{a_1(r^n - 1)}{r - 1}$$
where (a_1) is the first term, (r) is the common ratio, and (n) is the number of terms.
3. Sum of n terms of an arithmetic-geometric series.
$$S_n = \frac{a_1(r^n - 1)}{r - 1} - \frac{d(n - 1)}{2}$$
where (a_1) is the first term, (r) is the common ratio, (d) is the common difference, and (n) is the number of terms.
4. Sum of n terms of a harmonic series.
$$H_n = \sum\limits_{i=1}^n \frac{1}{i} \approx \ln n + \gamma$$
where (\gamma \approx 0.57721) is the Euler-Mascheroni constant.
5. Sum of n terms of a telescoping series.
Telescoping series are series of the form
$$\sum\limits_{i=1}^n (a_i - a_{i+1})$$
where (a_i) and (a_{i+1}) are consecutive terms of the series. The sum of a telescoping series can be found by simply subtracting the last term from the first term:
$$S_n = a_1 - a_{n+1}$$
6. Find the sum of n terms of the sequence {a_n} defined by a_n = 3n - 1.
$$S_n = \sum\limits_{i=1}^n (3i-1) = 3\cdot \frac{n(n+1)}{2} - n$$
7. Find the sum of n terms of the sequence {a_n} defined by a_n = 1/n.
$$S_n = \sum\limits_{i=1}^n \frac{1}{i} \approx \ln n + \gamma$$
8. Find the sum of n terms of the sequence {a_n} defined by a_n = n^2 - 1.
$$S_n = \sum\limits_{i=1}^n (i^2 - 1) = \frac{n(n+1)(2n+2)}{6}$$
