Vriti EdustoreSome properties & Shortcut for Series
Some useful properties of various type of Series (A.P , G.P , H.P )
1. Three numbers a,b,c are in A.P. if 2b = a+c i.e. twice the middle term = sum of extremes
2. If a constant is added/subtracted from each term of an A.P. then the resulting progression will also be an A.P.
Common difference of new A.P = k * Common difference of original A.P.
3. If each term of an A.P is multiplied by a non-zero constant, say k, then the resulting progression will also be an A.P.
4. If each term of an A.P. is divided by a non-zero constant, say k, then the resulting progression will also be an A.P.
Common difference of new A.P. = 1/k * Common difference of original A.P.
5. If the terms of an A.P are selected at regular intervals then, these selected terms also form an A.P. i.e. if a,b,c,d,e,f,g,h,i,j,k,.. form an A.P., then, a,d,g,j are also in A.P.
6. Consider a sequence containing n terms. Then mth term from end = (n-m+1)th term form beginning.
7. Consider an A.P. containing n terms.Then sum of the terms which are equidistant from the beginning & the end is a constant & equals to the sum of first & the last term i.e.
mth term from end + mth term from beginning = first term + last term = a + l
8. In an A.P, if Tp = q & Tq = p, then T(p+q) = 0 where Tr denotes rth term.
9. In an A.P., if p.Tp = q.Tq, then T(p+q) = 0
10. In an A.P. , if Sp = q & Sq = p, then S(p+q) = -(p+q), where, Sr denotes the sum of r terms
11. In an A.P. , if Sp = Sq , then S(p+q) = 0
12. Any three numbers in an A.P. may be selected as a-d,a,a+d.
13. Any four numbers in an A.P. may be selected as a-3d ,a-d, a+d, a+3d.
14. Any five numbeers in an A.P. may be selected as a-2d,a-d,a,a+d,a+2d.
15. If three numbers a, b, c are in G.P., then, their logarithms form an A.P. i.e. a,b,c form an G.P. if log a,log b, log c form an A.P.
16. If a,b,c form an A.P. , then x^a,x^b,x^c form a G.P.
17. Three numbers a, b, c are in G.P if b/a = c/b or b^2 = ac i.e. square of middle term = product of extremes.
18. If each term of a G.P is multiplied/divided by a non-zero constant, then the resulting progression will also be a G.P.
Common ratio of new G.P. = Common ratio of original G.P.
19. If each term of a G.P is raised to the same power, say k, then the resulting progression will also be a G.P.
Common ratio of new G.P = kth power of the common ratio of original G.P.
e.g. 1,2,4,8,16,32,64,... are in G.P. Cubing each term, we get, 1,8,64,512,... which form a G.P.
20. If the terms of a G.P. are selected at regular intervals then, these selected terms also form an G.P.
i.e. if a,b,c,d,e,f,g,h,i,j,k,....form a G.P, then, a,d,g,j are also in G.P.
21. Consider the sequence = k + kk + kkk + ....
Tn = nth term of above sequence = kkk....k = k/9(10^n - 1)
Sn = sum to n terms of above sequence
= 10k/81 (10^n – 1) – nk/9
22. Consider an G.P. containing n terms. Then the product of terms which are equidistant from the beginning & end is a constant & equals to the product of first & last term. i.e.
mth term from end * mth term from beginning = first term * last term = a * l
23. If nth term of a G.P. is k. Then , the product of first 2n – 1 terms of the G.P. is k^(2n-1).
24. If the nth term of a progression is a linear expression of n, i.e. Tn = an + b, then the progression will be an A.P. & its Common difference = a.
25. If the sum to n terms of a progression is a quadratic expression of n, i.e. Sn = an^2 + bc + c, then the progression will be an A.P. & its Common difference = 2a.
26. Any three numbers in an G.P. may be selected as a/r , a , ar .
27. Any four numbers in an G.P. may be selected as a/r^3, a/r, ar, ar^3 .
28. Any five numbers in an G.P. may be selected as a/r^2, a/r , a , ar , ar^2 .
29. Three numbers a, b, c are in H.P. if 1/a,1/b,1/c are in A.P i.e. b = 2ac/(a+c)
30. The three numbers X1,X2,X3,…..Xn are in H.P. if X1.X2 + X2.X3 + X3.X4 + …..Xn-1.Xn = (n-1).X1.Xn .
31. Let a, b, be the given numbers. Then,
Their Arithmetic mean i.e. AM = (a+b)/2
Their Geometric mean i.e. GM = 
Their Harmonic mean i.e. HM = 2ab/(a+b)
32. Let X1,X2,X3,X4,….Xn-1,Xn be the given numbers. Then,
Their Arithmetic Mean i.e. AM = ( X1+X2+X3+….+Xn )/ n
Their Geometric Mean i.e. GM = 
Their Harmonic Mean i.e. HM = 
33. If n Arithmetic means are intersected between two numbers a & b. Then,
Sum of n AM's = n[( a+b )/2 ]
34. If n Geometric means are inserted between two numbers a & b.Then ,
Product of n GM's = ( 
)^n
35. If A,G,H be the A.M, G.M. , H.M between two numbers a & b, then 
( sign of equality holds if a = b )
36. If A, G, H be the A.M,G.M,H.M between two numbers a & b, then G^2 = AH i.e.
A, G, H are in G.P.
37. If A, G, H be the A.M, G.M & H.M between two numbers a & b, then
(1) [ {a^(n+1) + b^(n+1)} / (a^n + b^n) ] = A , when n = 0
(2) [ {a^(n+1) + b^(n+1)} / (a^n + b^n) ] = G , when n = -1/2
(3) [ {a^(n+1) + b^(n+1)} / (a^n + b^n) ] = H , when n = -1
38. If 
then, 
where A & G denotes the A.M & G.M
between the numbers a & b.
39. If A & G denotes the A.M & G.M between the numbers a & b, then the numbers are the roots of the equation x^2 - 2Ax + G^2 = 0.
40. If A & G denotes the A.M & G.M between the numbers a & b, then the numbers are given by 
.
41. If A, G & H denotes the A.M,G.M & H.M respectively, of three numbers a, b & c, then the numbers are the roots of the cubic equation x^3 - 3AX^2 + 3G^3/H x - G^3 = 0.
42. Consider the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,....,where n consecutive terms have the value n.Then, to determine its kth term solve the following inequation :
Here, x = kth term
43. If X1,X2,X3,X4,X5,......Xn-1,Xn are the non-zero terms of a non-constant A.P, then
44. Let a, b, c are in A.P, p,q,r are in A.P & x,y,z also form an A.P . Then, the value of determinant
= (det)
= 0
i.e. if the element of each row(column) of a determinant form an A.P, then the value of the determinant will be zero.
45. If the sum of first n terms of two A.P's are in the ratio (an+b):(cn+d), then the ratio of their kth terms is given by (a(2k-1)+b):(c(2k-1)+d). i.e. to find the ratio of their kth terms, substitute n = 2k-1 in the given expression.
46. If X1,X2,X3,X4,.....Xn-1,Xn,Xn+1 are the non-zero terms of a non-constant A.P, whose common difference is d. Then
= 
47. If X1,X2,X3, & Y1,Y2,Y3, are terms of two A.P's. Then, the points (X1,Y1), (X2,Y2), (X3,Y3) will be collinear.
48. If X1,X2,X3, & Y1,Y2,Y3 are terms of two G.P's with the same common ratio. Then, the points (X1,Y1), (X2,Y2), (X3,Y3) will be collinear.
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