Progression and Series Formulas

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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 = \(\frac{b-a}{n+1}\), A₁ = a + d, A₂ = 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 + ar² + 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
G² = 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 G₁ = ar, G₂ = ar², G₃ = ar³ ……., G_n = arⁿ 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, ar²
• Four terms as: \(\frac{a}{r^{3}}, \frac{a}{r}\), ar, ar³

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

• General term of an A.G.P
  a, (a + d) r, (a + 2d) r² ,……. 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)
• G² = 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 = \(\sum_{k=1}^{n}(k)=\frac{n(n+1)}{2}\)
• 1² + 2² + 3² + ……. + n² = \(\sum_{k=1}^{n}(k)^{2}=\frac{n(n+1)(2 n+1)}{6}\)
• 1³ + 2³ + 3³ + ……. + n³ = \(\sum_{k=1}^{n}(k)^{3}=\left(\frac{n(n+1)}{2}\right)^{2}=\left(\sum_{k=1}^{n} k\right)^{2}\)

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