Advertisements
Advertisements
प्रश्न
If a, b, c are in G.P., then prove that:
Advertisements
उत्तर
\[\text{ a, b and c are in G . P }. \]
\[ \therefore b^2 = ac . . . . . . . . (i)\]
\[\text { Now, LHS }= \frac{a^2 + ab + b^2}{bc + ca + ab}\]
\[ = \frac{a^2 + ab + ac}{bc + b^2 + ab} \left[ \text { Using } (i) \right]\]
\[ = \frac{a\left( a + b + c \right)}{b\left( c + b + a \right)}\]
\[ = \frac{a}{b}\]
\[ = \frac{1}{r}\]
\[\text { Here, r = common ratio }\]
\[\text { RHS }= \frac{b + a}{c + b}\]
\[ = \frac{ar + a}{a r^2 + ar}\]
\[ = \frac{a(r + 1)}{ar(r + 1)}\]
\[ = \frac{1}{r}\]
\[ \therefore\text { LHS = RHS }\]
संबंधित प्रश्न
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Evaluate `sum_(k=1)^11 (2+3^k )`
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 `1/r^n`.
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
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, ...
Find the 4th term from the end of the G.P.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
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 the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if r = − 3 and t6 = 1701, find a.
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.
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
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, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`2.bar(4)`
Find : `sum_("r" = 1)^oo 4(0.5)^"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
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. 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 ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
