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
Advertisements
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
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
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?
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`.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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 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.
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:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
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:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
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:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
The fractional value of 2.357 is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For the G.P. if r = − 3 and t6 = 1701, find a.
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 10 years.
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then 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 ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... 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 ______.
