Advertisements
Advertisements
प्रश्न
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
उत्तर
\[\text { Since a, b, c are in A . P . , we have: } \]
\[2b = a + c\]
\[\text { We have to prove the following: } \]
\[2(ca - b^2 ) = \left( bc - a^2 + ab - c^2 \right)\]
\[\text { RHS }: bc - a^2 + ab - c^2 \]
\[ = c(b - c) + a(b - a)\]
\[ = c\left( \frac{a + c}{2} - c \right) + a\left( \frac{a + c}{2} - a \right) \left( \because 2b = a + c \right)\]
\[ = c\left( \frac{a + c - 2c}{2} \right) + a\left( \frac{a + c - 2a}{2} \right)\]
\[ = \frac{c\left( a - c \right)}{2} + a\left( \frac{c - a}{2} \right)\]
\[ = \frac{ca}{2} - \frac{c^2}{2} + \frac{ac}{2} - \frac{a^2}{2}\]
\[ = ac - \frac{1}{2}\left( c^2 + a^2 \right)\]
\[ = ac - \frac{1}{2}\left( 4 b^2 - 2ac \right) \left( \because a^2 + c^2 + 2ac = 4 b^2 \Rightarrow a^2 + c^2 = 4 b^2 - 2ac \right)\]
\[ = ac - 2 b^2 + ac\]
\[ = 2ac - 2 b^2 \]
\[ = 2\left( ac - b^2 \right)\]
\[ =\text { LHS }\]
\[\text { Hence, proved } . \]
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Find:
nth term of the A.P. 13, 8, 3, −2, ...
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
Write the common difference of an A.P. the sum of whose first n terms is
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is
If Sn denotes the sum of first n terms of an A.P. < an > such that
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
