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
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`
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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.
Let < an > be a sequence defined by a1 = 3 and, an = 3an − 1 + 2, for all n > 1
Find the first four terms of the sequence.
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
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.
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?
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all integers between 50 and 500 which are divisible by 7.
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.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
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.
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.
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 farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
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 ______.
The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.
Which type of A.P. is represented by the sequence \(2, 5, 8, 11, \ldots\)?
Which of the following is an example of a Constant A.P.?
If you save ₹500 in the first month, ₹700 in the second month, ₹900 in the third month, and so on, what is the common difference of the A.P. \(500, 700, 900, 1100, \ldots\)?
