Advertisements
Advertisements
प्रश्न
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
Advertisements
उत्तर
Given:
< an > is an A.P.
\[\frac{a_4}{a_7} = \frac{2}{3}\]
\[ \Rightarrow \frac{a + \left( 4 - 1 \right)d}{a + \left( 7 - 1 \right)d} = \frac{2}{3} \]
\[ \Rightarrow \frac{a + 3d}{a + 6d} = \frac{2}{3}\]
\[ \Rightarrow 3(a + 3d) = 2(a + 6d) \]
\[ \Rightarrow 3a + 9d = 2a + 12d\]
\[ \Rightarrow a = 3d . . . . (i)\]
\[\therefore \frac{a_6}{a_8} = \frac{a + \left( 6 - 1 \right)d}{a + \left( 8 - 1 \right)d}\]
\[ \Rightarrow \frac{a_6}{a_8} = \frac{a + 5d}{a + 7d}\]
\[ \Rightarrow \frac{a_6}{a_8} = \frac{3d + 5d}{3d + 7d} \left( \text { From }(i) \right)\]
\[ \Rightarrow \frac{a_6}{a_8} = \frac{8d}{10d}\]
\[ \Rightarrow \frac{a_6}{a_8} = \frac{4d}{5d} = \frac{4}{5}\]
APPEARS IN
संबंधित प्रश्न
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Find the sum of all numbers between 200 and 400 which are divisible by 7.
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
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the A.P. 3, 8, 13, ... is 248?
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
How many numbers of two digit are divisible by 3?
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.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
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).
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 a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also 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.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
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?
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
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))`.
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
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 ______.
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 n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 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 ______.
