Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
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 \[\frac{1}{r^n}\].
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 serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
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
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
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 Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The third term of G.P. is 4. The product of its first 5 terms is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
