Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let the first terms of the two A.P.'s be a and a'; and their common difference be d and d'.
Now,
\[\frac{S_n}{S_n '} = \frac{\left( 3n + 2 \right)}{\left( 2n + 3 \right)}\]
\[ \Rightarrow \frac{\frac{n}{2}\left[ 2a + \left( n - 1 \right)d \right]}{\frac{n}{2}\left[ 2a' + \left( n - 1 \right)d' \right]} = \frac{\left( 3n + 2 \right)}{\left( 2n + 3 \right)}\]
\[ \Rightarrow \frac{\left[ 2a + \left( n - 1 \right)d \right]}{\left[ 2a' + \left( n - 1 \right)d' \right]} = \frac{\left( 3n + 2 \right)}{\left( 2n + 3 \right)}\]
\[\text { Let }n = 23\]
\[ \Rightarrow \frac{2a + \left( 23 - 1 \right)d}{2a' + \left( 23 - 1 \right)d'} = \frac{3 \times 23 + 2}{2 \times 23 + 3}\]
\[ \Rightarrow \frac{2a + 22d}{2a' + 22d'} = \frac{69 + 2}{46 + 3}\]
\[ \Rightarrow \frac{2\left( a + 11d \right)}{2\left( a' + 11d' \right)} = \frac{71}{49}\]
\[ \therefore \frac{a_{12}}{a_{12'} } = \frac{71}{49}\]
So, the ratio of their 12th terms is 71 : 49.
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
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 ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Find:
nth term of the A.P. 13, 8, 3, −2, ...
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of first n odd natural numbers.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
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.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
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.
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?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
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
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))`.
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?
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
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).
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
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 ______.
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 ______.
