Advertisements
Advertisements
Question
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
Advertisements
Solution
a, b and c are in G.P.
\[\therefore b^2 = ac\] .......(1)
\[\text { LHS }= \frac{\left( a + b + c \right)^2}{a^2 + b^2 + c^2}\]
\[ = \frac{\left( a + b + c \right)^2}{a^2 - b^2 + c^2 + 2 b^2}\]
\[ = \frac{\left( a + b + c \right)^2}{a^2 - b^2 + c^2 + 2ac} \left[ \text { Using } (1) \right]\]
\[ = \frac{\left( a + b + c \right)^2}{\left( a + b + c \right)\left( a - b + c \right)} \left[ \because \left( a + b + c \right)\left( a - b + c \right) = a^2 - b^2 + c^2 + 2ac \right]\]
\[ = \frac{\left( a + b + c \right)}{\left( a - b + c \right)} =\text { RHS }\]
APPEARS IN
RELATED QUESTIONS
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Find the geometric means of the following pairs of number:
2 and 8
Find the geometric means of the following pairs of number:
−8 and −2
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
For the following G.P.s, find Sn
3, 6, 12, 24, ...
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
