z Let a be the first term, r be the common ratio and n
be the number of terms such that a, ar, ar^2 , ar^3 , .....
is a G.P., then
General term an = arn–1
Sum to n terms
Sar
rrar
rrnn= n−
−<−
−>⎧⎨⎪⎪⎩⎪
⎪(),(),1
111
11ififSum to infinite terms S a
r
∞= r
−<
1,| | 1GEOMETRIC MEAN (G.M.)
z Let a, b be any two numbers and G be the geometric
mean between them. Then a, G, b are in G.P.
⇒ Gab=
z Let a, b be any two numbers and G 1 , G 2 , ....., Gn be n
geometric means between them. Then a, G 1 , G 2 , .....,
Gn, b are in G.P.
⇒ =
⎛
⎝⎜⎞
⎠⎟
r b +
an1
1So,Gab
aGab
aGab
ann
nn
n
11
1
22
= ⎛^1
⎝⎜⎞
⎠⎟ =⎛
⎝⎜⎞
⎠⎟ =⎛
⎝⎜⎞
⎠⎟++, , ..... , + 11RELATIONSHIP BETWEEN A.M. AND G.M.Let a, b be any two numbers and let A=ab+ Gab=
2
,
Then, A ≥ G
SUM TO n TERMS OF SPECIAL SERIES123 1
2z (^) +++ +=..... n nn()+
123 12 1
6
z (^222) +++ +=..... n^2 nn()( )++n
123 1
2
333 3
2
+++ +=⎡ +
⎣⎢
⎤
⎦⎥
z ..... n nn()
PROBLEMS
Very Short Answer Type
- Find the sum of the series
99 + 95 + 91 + 87 + ..... to 20 terms. - Prove that the sum of n arithmetic means between
two numbers in n times the single A.M. between
them. - Find the sum of n terms of the series
(a + b) + (a^2 + 2b) + (a^3 + 3b)+ .....
4. If reciprocals of xy++y yz
22
,, are in A.P., show
that x, y, z are in G.P.- Find the sum to infinity of the G.P.
1 1
3
1
91
27+++ +.....Short Answer Type- Between two numbers whose sum is
13
6 an even^
number of A.M.’s are inserted. If the sum of means
exceeds their number by unity, find the number of
means.- If x, y, z are in A.P. and A 1 is the A.M. of x and y and
A 2 is the A.M. of y and z, then prove that the A.M.
of A 1 and A 2 is y. - The sum of three numbers in A.P. is –3 and their
product is 8. Find the numbers. - Find the sum of 10 terms of the G.P. 11
2
1
41
8, , , .....- If x > 0, prove that x+x≥2.
1Long Answer Type - I- If a^2 + 2bc, b^2 + 2ac, c^2 + 2ab are in A.P., show that
111
bccaab−− −
,, are in A.P.- If a, b, c, d are in G.P., show that
(b – c)^2 + (c – a)^2 + (d – b)^2 = (a – d)^2 - x + y + z =15 if a, x, y, z, b are in A.P. and
1115
xyz 3
++= if^11111
ax yzb,,,, are in A.P., find the
G.M. of a and b.- If the sum of n terms of a series is 5n^2 + 3n, find its
nth term. Are the term of this series in A.P.? - Find the sum of the series
- n + 2. (n – 1) + 3. (n − 2) + ..... + n. 1
Long Answer Type - II
- n + 2. (n – 1) + 3. (n − 2) + ..... + n. 1
- If a, b are the roots of equation x^2 – 3x + p = 0 and
c, d are roots of equation x^2 – 12x + q = 0, where a, b, c,
d forms a G.P., then prove that (q + p):(q – p) = 17 : 15. - If a, b, c are in A.P. and^111222
abc
,, are in A.P., then
prove that either −abc
2,, are in G.P. or a = b = c.