# Progression and Series Formulas

Are you fed up solving sums on Progression and Series and tired of doing lengthy calculations? Not anymore! Use our Progression and Series Formulas and make your work simple. Check out the List of Progressions and Series Formulae existing and get to know the concept behind them. Clarify your doubts on the concept as all of the related formulas are listed here.

## List of Progression and Series Formulae

Grab the opportunity and learn the Progression and Series Formulas prevailing and do your homework or assignments. With the list of Formulas on Progression and Series, you can get a good grip on the concept. Check out the basic and advanced formulas included in the concept of Progression & Series.

**1. Arithmetic Progression (A.P.)**

If a is the first term and d is the common difference then A.P. can be written as a + (a + d) + (a + 2d) + (a + 3d) + ………

**2. General term of an A.P.**

General term (n^{th} term) of an A.P. is given by T_{n} = a + (n – 1) d

**3. Sum of n terms of an A.P.**

S_{n} = \(\frac{n}{2}\)[2a + (n -1) d] or S_{n} = \(\frac{n}{2}\)[a + T_{n}]

(i) If sum of n terms S_{n} is given then general term T_{n} = S_{n} – S_{n-1} where S_{n-1} is sum of (n – 1) terms of A.P.

**4. Arithmetic Mean (A.M.)**

If A is the A.M. between two given numbers a and b, then

A = \(\frac{a+b}{2}\) ⇒ 2A = a + b

**5. n AM’s between two given numbers a and b**

d = \(\frac{b-a}{n+1}\), A_{1} = a + d, A_{2} = a + 2d,…. A_{n} = a + nd or A_{n} = b – d

and A_{r} = a + r \(\left(\frac{b-a}{n+1}\right)\) where A_{r} is r^{th} A.M. between a & b.

- Sum of n AM’s inserted between a and b is \(\frac{n}{2}\) (a + b)
- Any term of an A.P. (except the first term) is equal to the half of the sum of terms equidistant from the term i.e.

a_{n}= \(\frac{1}{2}\) (a_{n-k}+ a_{n+k}), k < n, where k is distance of term.

**6. Supposition of terms in A.P.**

- Three terms as : a – d, a, a + d
- Five terms as : a – 2d, a – d, a, a + d, a + 2d
- Four terms as : a – 3d, a – d, a + d, a + 3d

**7. Some standard results**

- Sum of first n natural numbers = \(\sum_{r=1}^{n} r=\frac{n(n+1)}{2}\)
- Sum of first n odd natural numbers = \(\sum_{r=1}^{n}(2 r-1)=n^{2} \)
- Sum of first n even natural numbers = \(\sum_{r=1}^{n} 2 r=n(n+1)\)
- Sum of squares of first n natural numbers = \(\sum_{r=1}^{n} r^{2}=\frac{n(n+1)(2 n+1)}{6}\)
- Sum of cubes of first n natural numbers = \(\sum_{r=1}^{n} r^{3}=\left[\frac{n(n+1)}{2}\right]^{2}\)
- If for an A.P., p
^{th}term is q, q^{th}term is p then m^{th}term is p + q – m. - If for an A.P., sum of p terms is q, sum of q terms is p, then sum of (p + q) terms is (p + q).
- If for an A.P., sum of p terms is equal to sum of q terms then sum of (p + q) terms is zero.

**8. General term of a G.P.**

General term (n^{th} term) of a G.P. a + ar + ar^{2} + is given by T_{n} = ar^{n-1}

**9. Sum of n terms of a G.P.**

The sum of first n terms of an G.P. is given by

S_{n} = \(\frac{a\left(1-r^{n}\right)}{1-r}=\frac{a-r T_{n}}{1-r}\) when r < 1 or S_{n} = \(\frac{a\left(r^{n}-1\right)}{r-1}=\frac{r T_{n}-a}{r-1}\)

when r > 1 and S_{n} = nr when r = 1

**10. Sum of an infinite G.P.**

The sum of an infinite G.P. with first term a and common ratio

r (- 1 < r < 1 i.e. |r| < 1) is S_{x} = \(\frac{a}{1-r}\)

**11. Geometrical Mean (G.M.)**

If G is the G.M. between two numbers a and b then

G^{2} = ab ⇒ G = \(\sqrt{a b}\)

**12. n GM’s between two given numbers a and b**

r = \(\left(\frac{b}{a}\right)^{\frac{1}{n+1}}\) then G_{1} = ar, G_{2} = ar^{2}, G_{3} = ar^{3} ……., G_{n} = ar^{n} G_{n} = \(\frac{b}{r}\) and G_{k} = a\(\left(\frac{b}{a}\right)^{\frac{k}{n+1}}\). where G_{k} is k^{th} G.M. between a & b.

product of n GM’s inserted between a and b is (ab)^{n/2}

**13. Supposition of terms in G.P.**

- Three terms as: \(\frac{a}{r}\), a, ar
- Five terms as: \(\frac{a}{r^{2}}, \frac{a}{r}\), a, ar, ar
^{2} - Four terms as: \(\frac{a}{r^{3}}, \frac{a}{r}\), ar, ar
^{3}

**14. Arithmetic – Geometrical Progression (A.G.P)**

- General term of an A.G.P

a, (a + d) r, (a + 2d) r^{2},……. is T_{n}= [a + (n – 1)d]r^{n-1} - Sum of n terms: S
_{n}= \(\frac{a}{1-r}+\frac{r \cdot d\left(1-r^{n-1}\right)}{(1-r)^{2}}-\frac{(a+(n-1) d) r^{n}}{(1-r)}\) - Sum of infinite terms S
_{∞}= \(\frac{a}{1-r}+\frac{d r}{(1-r)^{2}}\), |r| < 1

**15. General term of H.P.**

General term of an H.P. \(\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2 d}+\ldots . . \text { is } T_{n}=\frac{1}{a+(n-1) d}\)

NOTE: Sum of n terms in H.P. is not defined.

**16. Harmonical Mean (H.M.)**

If H is the H.M. between a and b then H = \(\frac{2 a b}{a+b}\)

**17. n H.M.’s between two given numbers**

To find n H.M.’s between a and b, we first find n A.M.’s between 1/a and 1/b then their reciprocals will be required H.M.’s.

**18. Relation between A.M., G.M. & H.M.**

If A, G, H are A.M., G.M. and H.M. between any two numbers

- A ≥ G ≥ H (equality holds when all terms are equal)
- G
^{2}= AH - If A and G are A.M., G.M. respectively between two positive numbers, then these numbers are A + \(\sqrt{A^{2}-G^{2}}\), A – \(\sqrt{A^{2}-G^{2}}\)

**19. Some special series:**

- 1 + 2 + 3 + n = \(\sum_{k=1}^{n}(k)=\frac{n(n+1)}{2}\)
- 1
^{2}+ 2^{2}+ 3^{2}+ ……. + n^{2}= \(\sum_{k=1}^{n}(k)^{2}=\frac{n(n+1)(2 n+1)}{6}\) - 1
^{3}+ 2^{3}+ 3^{3}+ ……. + n^{3}= \(\sum_{k=1}^{n}(k)^{3}=\left(\frac{n(n+1)}{2}\right)^{2}=\left(\sum_{k=1}^{n} k\right)^{2}\)