Advertisements
Advertisements
Question
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
Advertisements
Solution
a, b and c are in G.P.
\[\therefore b^2 = ac\] .......(1)
\[\text { LHS }= \left( a + 2b + 2c \right)\left( a - 2b + 2c \right)\]
\[ = a^2 - 4 b^2 + 4 c^2 + 4ac\]
\[ = a^2 - 4ac + 4 c^2 + 4ac \left[ \text { Using }(1) \right]\]
\[ = a^2 + 4 c^2 = \text { RHS }\]
RELATED QUESTIONS
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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 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 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the rational number whose decimal expansion is `0.4bar23`.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
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 the following is also in G.P.:
a3, b3, c3
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, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A 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
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
The numbers 3, x, and x + 6 form are in G.P. Find x
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
For a G.P. If t3 = 20 , t6 = 160 , find S7
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,...`
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
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
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:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
The third term of G.P. is 4. The product of its first 5 terms is ______.
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 ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
