Advertisements
Advertisements
Question
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
Advertisements
Solution
a, b and c are in G.P.
∴ \[b^2 = ac . . . . . . . (1)\]
\[\left( ab + bc \right)^2 = \left( ab \right)^2 + 2a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( ab \right)^2 + a b^2 c + a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 b^2 + ac\left( ac \right) + b^2 \left( b^2 \right) + b^2 c^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 \left( b^2 + c^2 \right) + b^2 \left( b^2 + c^2 \right)\]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( b^2 + c^2 \right)\left( a^2 + b^2 \right)\]
\[\text { Therefore }, \left( a^2 + b^2 \right), \left( b^2 + c^2 \right) \text { and }\left( ab + bc \right) \text { are also in G . P } . \]
RELATED QUESTIONS
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.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
Mark the correct alternative in the following question:
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 p2R3 : S3 is equal to
Which term of the G.P. 5, 25, 125, 625, … is 510?
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
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
For a G.P. a = 2, r = `-2/3`, find S6
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
