Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text {As, x, y and z are in A . P } . \]
\[\text { So, y } = \frac{x + z}{2} . . . . . \left( i \right)\]
\[\text { Now }, \]
\[\left( x^2 + xy + y^2 \right) + \left( y^2 + yz + z^2 \right)\]
\[ = x^2 + z^2 + 2 y^2 + xy + yz\]
\[ = x^2 + z^2 + 2 y^2 + y\left( x + z \right)\]
\[ = x^2 + z^2 + 2 \left( \frac{x + z}{2} \right)^2 + \left( \frac{x + z}{2} \right)\left( x + z \right) \left[ \text { Using } \left( i \right) \right]\]
\[ = x^2 + z^2 + 2\left( \frac{\left( x + z \right)^2}{4} \right) + \frac{\left( x + z \right)^2}{2}\]
\[ = x^2 + z^2 + \frac{\left( x + z \right)^2}{2} + \frac{\left( x + z \right)^2}{2}\]
\[ = x^2 + z^2 + \left( x + z \right)^2 \]
\[ = x^2 + z^2 + x^2 + 2xy + z^2 \]
\[ = 2 x^2 + 2xy + 2 z^2 \]
\[ = 2\left( x^2 + xy + z^2 \right)\]
\[\text { Since, } \left( x^2 + xy + y^2 \right) + \left( y^2 + yz + z^2 \right) = 2\left( x^2 + xy + z^2 \right)\]
\[\text { So, } \left( x^2 + xy + y^2 \right), \left( x^2 + xy + z^2 \right) \text { and } \left( y^2 + yz + z^2 \right) \text { are in A . P } .\]
Hence, x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P.
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
Is 68 a term of the A.P. 7, 10, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
How many numbers of two digit are divisible by 3?
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of first n natural numbers.
Find the sum of all integers between 50 and 500 which are divisible by 7.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If Sn denotes the sum of first n terms of an A.P. < an > such that
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is ______.
