Advertisements
Advertisements
Question
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
Solution
\[\text {As, x, y and z are in A . P } . \]
\[\text { So, y } = \frac{x + z}{2} . . . . . \left( i \right)\]
\[\text { Now }, \]
\[\left( x^2 + xy + y^2 \right) + \left( y^2 + yz + z^2 \right)\]
\[ = x^2 + z^2 + 2 y^2 + xy + yz\]
\[ = x^2 + z^2 + 2 y^2 + y\left( x + z \right)\]
\[ = x^2 + z^2 + 2 \left( \frac{x + z}{2} \right)^2 + \left( \frac{x + z}{2} \right)\left( x + z \right) \left[ \text { Using } \left( i \right) \right]\]
\[ = x^2 + z^2 + 2\left( \frac{\left( x + z \right)^2}{4} \right) + \frac{\left( x + z \right)^2}{2}\]
\[ = x^2 + z^2 + \frac{\left( x + z \right)^2}{2} + \frac{\left( x + z \right)^2}{2}\]
\[ = x^2 + z^2 + \left( x + z \right)^2 \]
\[ = x^2 + z^2 + x^2 + 2xy + z^2 \]
\[ = 2 x^2 + 2xy + 2 z^2 \]
\[ = 2\left( x^2 + xy + z^2 \right)\]
\[\text { Since, } \left( x^2 + xy + y^2 \right) + \left( y^2 + yz + z^2 \right) = 2\left( x^2 + xy + z^2 \right)\]
\[\text { So, } \left( x^2 + xy + y^2 \right), \left( x^2 + xy + z^2 \right) \text { and } \left( y^2 + yz + z^2 \right) \text { are in A . P } .\]
Hence, x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P.
APPEARS IN
RELATED QUESTIONS
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
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 numbers between 200 and 400 which are divisible by 7.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
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?
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
Is 68 a term of the A.P. 7, 10, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of first n odd natural numbers.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Find the sum of all even integers between 101 and 999.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
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
If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
