Advertisements
Advertisements
Question
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
Advertisements
Solution
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[ad = bc \]
\[ c^2 = bd\] .......(1)
\[\left( ab + bc + cd \right)^2 = \left( ab \right)^2 + \left( bc \right)^2 + \left( cd \right)^2 + 2a b^2 c + 2b c^2 d + 2abcd\]
\[ \Rightarrow \left( ab + bc + cd \right)^2 = a^2 b^2 + b^2 c^2 + c^2 d^2 + a b^2 c + a b^2 c + b c^2 d + b c^2 d + abcd + abcd\]
\[ \Rightarrow \left( ab + bc + cd \right)^2 = a^2 b^2 + b^2 c^2 + c^2 d^2 + b^2 \left( b^2 \right) + ac\left( ac \right) + c^2 \left( c^2 \right) + bd\left( bd \right) + bc\left( bc \right) + ad\left( ad \right) \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( ab + bc + cd \right)^2 = a^2 b^2 + a^2 c^2 + a^2 d^2 + b^4 + b^2 c^2 + b^2 d^2 + c^2 b^2 + c^4 + c^2 d^2 \]
\[ \Rightarrow \left( ab + bc + cd \right)^2 = a^2 \left( b^2 + c^2 + d^2 \right) + b^2 \left( b^2 + c^2 + d^2 \right) + c^2 \left( b^2 + c^2 + d^2 \right)\]
\[ \Rightarrow \left( ab + bc + cd \right)^2 = \left( b^2 + c^2 + d^2 \right)\left( a^2 + b^2 + c^2 \right)\]
\[\text { Therefore, }\left( a^2 + b^2 + c^2 \right), \left( ab + bc + cd \right) \text{ and }\left( b^2 + c^2 + d^2 \right) \text {are also in G . P } .\]
APPEARS IN
RELATED QUESTIONS
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
a3b and ab3
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
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
0.7, 0.07, 0.007, .....
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
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
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:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
