Advertisements
Advertisements
प्रश्न
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
उत्तर
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[bc = ad\]
\[ c^2 = bd\] .......(1)
\[\text { LHS }= \left( a + b + c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( a + b \right)\left( c + d \right) + \left( c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( ac + ad + bc + bd \right) + \left( c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( b^2 + bc + bc + c^2 \right) + \left( c + d \right)^2 \left[ \text { Using } (1) \right]\]
\[ = \left( a + b \right)^2 + 2 \left( b + c \right)^2 + \left( c + d \right)^2 = \text { RHS }\]
APPEARS IN
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
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.
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.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Find the sum of the following serie:
5 + 55 + 555 + ... 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:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
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 (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5thhour?
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
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.
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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 –
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
The third term of G.P. is 4. The product of its first 5 terms is ______.
