Advertisements
Advertisements
प्रश्न
If Sn denotes the sum of first n terms of an A.P. < an > such that
पर्याय
\[\frac{2 m + 1}{2 n + 1}\]
\[\frac{2 m - 1}{2 n - 1}\]
\[\frac{m - 1}{n - 1}\]
\[\frac{m + 1}{n + 1}\]
उत्तर
\[\frac{2 m - 1}{2 n - 1}\]
\[\frac{S_m}{S_n} = \frac{m^2}{n^2}\]
\[ \Rightarrow \frac{\frac{m}{2}\left\{ 2a + \left( m - 1 \right)d \right\}}{\frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}} = \frac{m^2}{n^2}\]
\[ \Rightarrow \frac{\left\{ 2a + \left( m - 1 \right)d \right\}}{\left\{ 2a + \left( n - 1 \right)d \right\}} = \frac{m}{n} \]
\[ \Rightarrow 2an + ndm - nd = 2am + nmd - md\]
\[ \Rightarrow 2an - 2am - nd + md = 0\]
\[ \Rightarrow 2a\left( n - m \right) - d\left( n - m \right) = 0\]
\[ \Rightarrow 2a\left( n - m \right) = d\left( n - m \right)\]
\[ \Rightarrow d = 2a . . . . . \left( 1 \right)\]
\[\text { Ratio of } \frac{a_m}{a_n} = \frac{a + \left( m - 1 \right)d}{a + \left( n - 1 \right)d}\]
\[ \Rightarrow \frac{a_m}{a_n} = \frac{a + \left( m - \right)2a}{a + \left( n - \right)2a}\text { From } (1)\]
\[ = \frac{a + 2am - 2a}{a + 2an - 2a} \]
\[ = \frac{2am - a}{2an - a}\]
\[ = \frac{2m - 1}{2n - 1}\]
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
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?
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
Which term of the A.P. 3, 8, 13, ... is 248?
Which term of the A.P. 4, 9, 14, ... is 254?
Is 68 a term of the A.P. 7, 10, 13, ...?
Is 302 a term of the A.P. 3, 8, 13, ...?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
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.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
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 natural numbers.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
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 a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
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.
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 sum of first n even natural numbers.
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.
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum 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
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))`.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
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 ______.
