RADIUS JEE

The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .................}
(G) Sum of n terms of an Arithmetic Series
Let an AP has n terms
first term = a
Last term = l = a + (n -1)d
Sum, Sn = n/2( a + l)
Sn = n/2( 2a+(n-1)d)
(H) Important Note
1. The sum of the infinite arithmetic sequence does not exist.
2. Sum of first n natural numbers
= n(n+1)/2
3. Sum of first n odd natural number = n^2
4. Sum of first n even natural numbers = n(n+1)
5. Sum of Square of first n natural numbers = n(n+1)(2n+1)/6
6. Sum of Cubes of first n natural numbers = [ n(n+1)/2]^2
7. sum of the fourth power of n natural numbers
= n(n +1)(2n +1)( 3n^2 +3n -1)/30
8. If rth term of an AP is
Tr = Ar^3 + Br^2 + Cr + D
then
Sum of n terms of that AP will be
Sn = Summation r =1 to r =n (Tr)
9. If in an AP pth term is q and qth term is p , then rth term will be = p + q - r
10. If in an AP sum of p terms is q and sum of q terms is p so, sum of (p+q) terms will be = -(p+q)
11. If in an AP sum of p terms is equal to sum of q terms then sum of (p +q) term will be Zero (0)
12. For an Arithmetic Series if sum of n terms given as Sn so we can calculate nth term Tn as Tn = Sn - Sn-1
13. Common Difference(d) of an AP can be calculated by d = S2 - 2S1
where S2 = Sum of first 2 terms of that AP
S1 = Sum of First term of that AP

14. Sum of infinite Arithmetic progression is infinity if d is > 0 and Sum of Infinite Arithmetic progression - Infinity if d is <0

15. If sum of n terms of an AP is in the form
Sn = An^2 + Bn +C ( Quadratic form )
So Common Difference(d) will be = 2 × Coefficient of n ^2
d = 2A

16. If nth term Tn of an AP is in the form of
Tn = Pn + Q ( Linear Expression )
Common difference(d) = Coefficient of n = P
17. Let there are two AP, AP1 and AP2
Let sum of n terms of AP1 = S1n = f1(n)
Let sum of n terms of AP2 = S2n = f2(n)
And if
S1n/S2n = f1(n)/f2(n)
Then
T1(n)/T2(n) = f1(2n-1)/f2(2n-1)
Here
T1n = nth term of AP1
T2n = nth term of AP2

18. Let if T1n/T2n = f1(n)/f2(n)
Then S1n / S2n = f1[(n+1)/2]/f2[(n+1)/2]

(I) Arithmetic Mean
(1) Arithmetic mean between two numbers p and q will be AM = (p +q)/2
p , AM, q .... are in AP
AM - p = q - AM
2AM = p + q
AM = (p +q)/2
(2) Arithmetic mean of n numbers a1, a2, a3, .....an
AM = (a1 + a2 + a3 + .... an)/n

(J) n Arithmetic Mean between two numbers p and q
(1)
p, A1, A2, A3, ........ An, q are in AP
First term(a)= p
Total number of term(N) = n +2
Last term(Tn) = q
So,
Tn = a + (N-1)d
d = Common Difference
q = p + (n +2 -1)d
q - p = (n+1)d
d = (q-p)/(n+1)

A1 = p + d = p + (q-p)/(n+1)
A2 = p + 2d = p + 2(q-p)/(n+1)

.........
...
..
An = p + nd = p + n(q-p)/(n+1) = q - d
(2) Sum of n AM between p and q
A1 + A2 + A3 .....+ An = n(p+q)/2
(3) Sum of r AM between p and q
A1 + A2 + A3 ...... + Ar = r(p+q)/2
(4) Sum of n AM between p and q / Sum of r AM between p and q = [n(p+q)/2]/[r(p+q)/2] = n/r
(5) Select 3 terms in AP
a - d , a , a + d
(6) Select 5 terms in AP
a - 2d, a - d, a, a +d, a + 2d
(7) Select 7 terms in AP
a - 3d, a - 2d, a - d, a, a +d, a + 2d, a + 3d
(8) Select 4 terms in AP
a - 3d, a - d, a + d, a + 3d

(9) Select 6 terms in AP
a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
(10) Select 8 terms in AP
a - 7d, a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d, a + 7d
(11) For a series, if mth term in the form of T(m) = Cm + D, Then this Series is in Arithmetic Progression
(12) For a series if sum of m terms in the form of S(m) = Cm^2 + Dm + E, then this Series will be in AP
(13) T(m) = Cm + D, = mth term of a Series ( AP )
Then T(m/k), T(mk), T(m+k), T(m-k) ..Will be in AP k is a non zero constant

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1.33K viewsGyanesh Rai, 15:35