Advertisements
Advertisements
प्रश्न
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
Advertisements
उत्तर
Given:
\[S_n = 3 n^2 \]
\[\text { For } n = 1, S_1 = 3 \times 1^2 = 3\]
\[\text { For } n = 2, S_2 = 3 \times 2^2 = 12\]
\[\text { For } n = 3, S_3 = 3 \times 3^2 = 27 \]
\[\text { and so on }\]
\[ \therefore S_1 = a_1 = 3\]
\[ a_2 = S_2 - S_1 = 12 - 3 = 9\]
\[ a_3 = S_3 - S_2 = 27 - 12 = 15\]
\[\text { and so on }\]
\[\text { Thus, the A . P . is } 3, 9, 15 . . . \]
APPEARS IN
संबंधित प्रश्न
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`
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
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 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
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the A.P. 4, 9, 14, ... is 254?
Is 68 a term of the A.P. 7, 10, 13, ...?
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
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.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of all integers between 100 and 550, which are divisible by 9.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
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.
Find the sum of odd integers from 1 to 2001.
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.
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).
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th 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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
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?
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
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 p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q 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 a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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))`.
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
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 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 ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.
