## Sequences and Series

# Arithmetic Progression (A.P.)

**Tips :**

- If an A.P has ‘n’ terms, then the n
^{th}term is called the last term of A.P and it is denoted by l. That is l = a+ (n – 1)d - Three numbers a, b, c are in A.P if and only if b – a = c – b
- If ‘a’ is the first term and ‘d’ is the common difference of an A.P having ‘m’ terms, then n
^{th}term from the end is (m – n + 1)^{th}term from the beginning. Thus n^{th}term from the end is a+(m – n)d - Any ‘3’ numbers in an A.P can be a – d, a, a + d
- Sum of ‘n’ terms of an A.P is \tt S_{n}= \frac{n}{2}\left[2a+\left(n-1\right)d\right]
- If a
_{1}, a_{2}, a_{3}...... a_{n}are in A.P then a_{1}+ k, a_{2}+ k, a_{3}+ k ....... are also in A.P. - If a
_{1}, a_{2}, a_{3}....... a_{n}are in A.P then a_{1}– k, a_{2}– k, a_{3}– k ....... are also in A.P. - If a
_{1}, a_{2}, a_{3}........ a_{n}are in A.P then ka_{1}, ka_{2,}ka_{3}........ ka_{n}are also in A.P. - If a
_{1}, a_{2}, a_{3}....... a_{n}are in A.P then \tt \frac{a_{1}}{k},\frac{a_{2}}{k},\frac{a_{3}}{k} ...... \left(k \neq 0 \right) are also in A.P. - If a
_{1}, a_{2}, a_{3}......... and b_{1}, b_{2}, b_{3}.......... are two A.P.s. then a_{1}+ b_{1}, a_{2}+ b_{2}, a_{3}+ b_{3}....... are also in A.P. other operations ‘ –‘ and ‘x’ will satisfy this. - If a
_{1}, a_{2}, a_{3}.......... and b_{1}, b_{2}, b_{3}............ are two A.P.s. then \tt \frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\frac{a_{3}}{b_{3}} ........ may not be in A.P. - If a
_{1}, a_{2}, a_{3}....... a_{n}are in A.P then a_{1}+ a_{n}= a_{2}+ a_{n-1}= a_{3}+ a_{n-2}= ........... - If a
_{1}, a_{2}, a_{3}............ a_{n}are in A.P then \tt a^{r}=\frac{a_{r-k} + a_{r + k}}{2} , 0 \leq k \leq n-r , - If n
^{th}term of a sequences is a linear expression in ‘n’ then the sequence is an A.P - If the sum of first ‘n’ terms of sequences is a quadratic expression in ‘n’, then the sequence is an A.P
- If a
_{1}, a_{2}, a_{3}, a_{4}........ a_{n}are in A.P then terms taken at regular intervals from this A.P are also in A.P. Ex : a_{1}, a_{4}, a_{7}, a_{10}..... also form an A.P - Arithmetic mean of ‘a’ and ‘b’ is \tt \frac{a+b}{2}
- \tt d=\frac{b-a}{n+1} (d is common difference) a is first term ‘b’ is last term.

**Tricks :**

- If a
_{n}−a_{n−1}is independent of n, the given sequence is an A.P. Otherwise it is not an A.P - If S
_{n}is the sum of ‘n’ terms of an A.P whose first term is ‘a’ and last term is ‘l’, then \tt S_{n}=\frac{n}{2}\left(a + l\right). - If common difference d, number of terms ‘n’ and the last term is ‘l’, are given then \tt S_{n}=\frac{n}{2}\left[2 l -\left(n-1\right)d\right]
- t
_{n}= S_{n}− S_{n−1 } - The sum of ‘n’ arithmetic means between two given numbers is ‘n’ times the single A.M between them, i.e if a and b are '2' given number of and A
_{1}, A_{2}, ....... A_{n}are 'n' arithmetic means between them, then \tt A_{1}+A_{2}+......A_{n}=n\frac{\left(a+b\right)}{2} - Sum to n terms of the series of the form (t
_{1}t_{2}......... t_{k})+ (t_{2}t_{3}........ t_{k+1})+ .......+ (t_{n}t_{n+1}.......... t_{n+k-1}) is \tt S_{n}=\frac{1}{\left(k+1\right)\left(t_{2}-t_{1}\right)} (t_{n}·t_{n+1}....... t_{n+k}..... to t_{1}t_{2}.......... t_{k})

### Part1: View the Topic in this video From 37:10 To 49:08

### Part2: View the Topic in this video From 00:40 To 42:16

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. Let 'a' be the first term and 'd' be the common difference of an A.P. Then its n^{th} term is a + (n − 1)d i.e., **T _{n} = a + (n **

**− 1)d**.

2. The sum of n terms of the series a + (a + d) + (a + 2d) + ...... + {a + (n − 1)d} is given by

\tt S_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]

Also, \tt S_{n}=\frac{n}{2}\left(a+l\right), where l = last term = a + (n − 1) d.

3. If a and b are two real numbers then single A.M. between a and b = \tt \frac{a+b}{2}

4. The sum of terms of an A.P. equidistant from the beginning and the end is constant and is equal to sum of first and last term. i.e. a

_{1}+ a

_{n}= a

_{2}+ a

_{n−1}= a

_{3}+ a

_{n}

_{−2}= ....

5. Any term of an A.P. (except the first and the last) is equal to the half the sum of terms which are equidistant from it. i.e. a_{n}=\frac{1}{2}(a_{n-k}+a_{n+k}) where k < n.

6. Three numbers a, b, c are in A.P. iff 2b = a + c.

7. If T

_{n}, T

_{n+1}and T

_{n+2}are three consecutive terms of an A.P., then 2T

_{n+1}= T

_{n}+ T

_{n+2}.

8. T

_{n}= S

_{n}− S

_{n}

_{−1}(n ≥ 2).