English

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
Advertisements

Question

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

Solution

a, b, c and d are in G.P.

\[\therefore b^2 = ac\]

\[bc = ad\]

\[ c^2 = bd\]             .......(1)

\[\text { LHS }= \left( a + b + c + d \right)^2 \]

\[ = \left( a + b \right)^2 + 2\left( a + b \right)\left( c + d \right) + \left( c + d \right)^2 \]

\[ = \left( a + b \right)^2 + 2\left( ac + ad + bc + bd \right) + \left( c + d \right)^2 \]

\[ = \left( a + b \right)^2 + 2\left( b^2 + bc + bc + c^2 \right) + \left( c + d \right)^2 \left[ \text { Using } (1) \right]\]

\[ = \left( a + b \right)^2 + 2 \left( b + c \right)^2 + \left( c + d \right)^2 = \text { RHS }\]

shaalaa.com
  Is there an error in this question or solution?
Chapter 20: Geometric Progression - Exercise 20.5 [Page 46]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 20 Geometric Progression
Exercise 20.5 | Q 9.2 | Page 46

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`


The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.


Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…


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.


Show that one of the following progression is a G.P. Also, find the common ratio in case:

\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]


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, ...


The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th 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.


If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].


Find the sum of the following geometric progression:

1, 3, 9, 27, ... to 8 terms;


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 \[\frac{1}{r^n}\].


How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?


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.


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.


Find the rational numbers having the following decimal expansion: 

\[3 . 5\overline 2\]


If a, b, c are in G.P., prove that log a, log b, log c are in A.P.


Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.


If a, b, c are in G.P., prove that:

\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]


If a, b, c, d are in G.P., prove that:

(b + c) (b + d) = (c + a) (c + d)


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 are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.


Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .


If the fifth term of a G.P. is 2, then write the product of its 9 terms.


If logxa, ax/2 and logb x are in G.P., then write the value of x.


The value of 91/3 . 91/9 . 91/27 ... upto inf, is 


If pq be two A.M.'s and G be one G.M. between two numbers, then G2


The numbers 3, x, and x + 6 form are in G.P. Find nth 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 a G.P. If t3 = 20 , t6 = 160 , find S7


If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]


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, ...`


Find : `sum_("n" = 1)^oo 0.4^"n"`


Find GM of two positive numbers whose A.M. and H.M. are 75 and 48


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:

Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.


Answer the following:

Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...


Answer the following:

If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2   


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×