Advertisements
Advertisements
प्रश्न
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
विकल्प
10
12
13
14
Advertisements
उत्तर
14
The given series is 1, . . . . . . . . . . . , 31
There are n A.M.s between 1 and 31:
\[1, A_1 , A_2 , A_3 , . . . . . , A_n , 31\]
Common difference, d = \[\frac{31 - 1}{n + 1} = \frac{30}{n + 1}\]
Here, we have:
\[\frac{A_1}{A_n} = \frac{3}{29}\]
\[ \Rightarrow \frac{1 + d}{1 + nd} = \frac{3}{29}\]
\[ \Rightarrow \frac{1 + \frac{30}{n + 1}}{1 + n \times \frac{30}{n + 1}} = \frac{3}{29}\]
\[ \Rightarrow \frac{n + 1 + 30}{n + 1 + 30n} = \frac{3}{29}\]
\[ \Rightarrow \frac{n + 31}{31n + 1} = \frac{3}{29}\]
\[ \Rightarrow 29n + 899 = 93n + 3\]
\[ \Rightarrow 64n = 896\]
\[ \Rightarrow n = 14\]
संबंधित प्रश्न
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
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?
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 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
Which term of the A.P. 84, 80, 76,... is 0?
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.
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 all natural numbers between 1 and 100, which are divisible by 2 or 5.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th 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.
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.
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.
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th 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 the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
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 a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
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?
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
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 ______.
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 ______.
