1. Arithmetic progression (A.P.):
(a) General A.P.-a,a+d,a+2d, …. A + (n-1) d where a is the first term add d is the common difference
(b) General (nth) term of an A.P. –
Tn = a + (n-1) d [nth term form the beginning ]If an A.P. having m term, term, then nth term form end = a+ (m-n)d
(c) sum of n terms of an A.P.-
Note: If sum of n term i.e. Sn is given then Tn = Sn - Sn-1 where Sn-1 is sum of (n-1) term
(d) Supposition of terms in A.P.—
(i) there term as a – d, a+d
(ii) four terms as a -3d, a-d, a+d,a+3d
(iii) five term terms as a -2d,a-d,a,a+d,a+2d
(e) Arithmetic mean (A.M.):
(i) A.M. of n numbers A1,A2 , …… An is defined as
(ii) For an A.P., A.M. of the terms taken symmetrically fprm the beginning and form the end will always be constant and will be equal to middle term or A.M. of middle term .
(iii) If A is the A.M. between two given nos. a and b, then
(iv) If A1,A2, ….. An are n A.M’ s between a and b, then A = A+d ,A2= a+2d , ….. An= a+nd, where d =(b-a)/(n+1)
(v) Sum of v A.M’s inserted between a and b is (n/2) (a+b)
(vi) Any term of an A.P. (except first term ) is equal to the half of the sum of term equidistant form the term i.e. an = 1/2 (an-r+an+r), r
2. Geometric Progression (G.P)
(a) General G.P. – a, ar , ar2 , …… where a is the first term and r is the common ratio
(b) General (nth) term of G.P. - -Tn = arn-1 If a G.P. having m term then nth term form end = arm-n
(c) Sum of n term of a G.P. ---
(d) Sum of an infinite G.P. --- S_ = (a/(1-r)) , |r| < 1
(e) Supposition of term in G.P. ---
(i) There term as (a/r),a,ar
(ii) Four term as (a/r3) , (a/r) ar, ar3
(iii) Five term as (a/r2),(a/r) ,a,ar.ar3
(f) Geometric Mean (G.M.) –
(i) Geometrical mean of n numbers x1, x2, …… xn is defined as G.M. = (x1x2 …… xn)1/n
(i) If G is the G.M. between two given number a and b, then
G2 = ab Þ G = Öab
(ii) If G1, G2 , …… Gn are n G.M’s between a and b, then
G1 = ar , G2 = ar2 ,…… Gn = arn, where r = (b/a)1/n+1
(iii) Product of the n G.M. ‘s inserted between a & b is (ab)n/2
3. Arithmetico – Geometric progression (A.G.P.):
(a) General form – a, (a+d) r, (a+2d)r2, …….
(b) General (nth) term – Tn = [a+(n-1) d] rn-1
(c) Sum of n term of an A.G.P – Sn
(d) Sum of infinite term of an A.G.P
4. Sum standerd result:
(a) ån = 1+2+3+ ….. + n
(b) ån2 = 12 + 22 + 32 + ….. + n2
(c) ån3 = 13 + 22 + 33 + ……. + n3
(d) åa = a+a+ …. + (n times) = na
(e) å(2n-1) = 1+3+5+ ….. (2n-1) = n2
(f) å2n = 2+4+6+ …… + 2n = n(n+1)
5. Harmonic Progression (H.P)
(c) Harmonic Mean (H.M.)
(i) If H is H.M. between a and b , then H = (2ab/(a+b))
(ii) If H1, H2 , …….., Hn are n H.M’s between a nad b ,
then H1 H1 = (ab(n+1) / (bn+a)) ,……… Hn = (ab(n+1) / (na+b)) or first n A.M.’s between (1/a) & (1/b) , then their reciprocal will be required H.M’s
6. Relation Between A.M. , G.M. and H.M.
(i) AH = G2
(ii) A ³ G ³
(iii) If A and G are A.M. and G.M. respectively between two + ve numbers , then these number are
A ± Ö(A2 -G2)
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