Advertisements
Advertisements
प्रश्न
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Advertisements
उत्तर
\[\text { Let the numbers be } (a - d), a, (a + d) . \]
\[\text { Sum } = a - d + a + a + d = 12\]
\[ \Rightarrow 3a = 12\]
\[ \Rightarrow a = 4\]
\[\text { Also }, (a - d )^3 + a^3 + (a + d )^3 = 288\]
\[ \Rightarrow a^3 - d^3 - 3 a^2 d + 3a d^2 + a^3 + a^3 + d^3 + 3 a^2 d + 3a d^2 = 288\]
\[ \Rightarrow 3 a^3 + 6a d^2 = 288\]
\[ \Rightarrow 3 \left( 4 \right)^3 + 6 \times 4 \times d^2 = 288\]
\[ \Rightarrow 192 + 24 d^2 = 288\]
\[ \Rightarrow 24 d^2 = 96\]
\[ \Rightarrow d^2 = 4\]
\[ \Rightarrow d = \pm 2\]
\[\text { When a = 4, d = 2, the numbers are } 2, 4, 6 . \]
\[\text { When a = 4, d = - 2, the numbers are } 6, 4, 2 .\]
संबंधित प्रश्न
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.
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
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
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.
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.
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?
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of first n odd natural numbers.
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
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 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an 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 man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
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?
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
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
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.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
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 ______.
