Advertisements
Advertisements
प्रश्न
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
उत्तर
a, b and c are in G.P.
\[\therefore b^2 = ac\] .......(1)
\[\text { LHS }= \left( a + 2b + 2c \right)\left( a - 2b + 2c \right)\]
\[ = a^2 - 4 b^2 + 4 c^2 + 4ac\]
\[ = a^2 - 4ac + 4 c^2 + 4ac \left[ \text { Using }(1) \right]\]
\[ = a^2 + 4 c^2 = \text { RHS }\]
APPEARS IN
संबंधित प्रश्न
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
If the pth , qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Which term of the G.P. :
\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]
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.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the 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.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
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 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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.
Find the geometric means of the following pairs of number:
a3b and ab3
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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
The nth term of a G.P. is 128 and the sum of its n terms is 225. 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
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.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
For the G.P. if a = `7/243`, r = 3 find t6.
For a G.P. If t4 = 16, t9 = 512, find S10
Express the following recurring decimal as a rational number:
`2.bar(4)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
If the first term of the G.P. is 6 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
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 ______.
