Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given:
\[\frac{S_n}{{S_n}^1} = \frac{2n + 5}{3n + 4}\]
\[ \Rightarrow \frac{\frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}}{\frac{n}{2}\left\{ 2b + \left( n - 1 \right) d^1 \right\}} = \frac{2n + 5}{3n + 4}\]
\[ \Rightarrow \frac{2a + \left( n - 1 \right)d}{2b + \left( n - 1 \right) d^1} = \frac{2n + 5}{3n + 4} . . . \left( 1 \right)\]
Ratio of their m terms =\[\frac{a_m}{b_m}\]
To find the ratio of the mth terms, replace n by 2m \[-\] 1 in equation (1).
\[\Rightarrow \frac{2a + \left( 2m - 2 \right)d}{2b + \left( 2m - 2 \right) d^1} = \frac{2\left( 2m - 1 \right) + 5}{3\left( 2m - 1 \right) + 4}\]
\[ \Rightarrow \frac{a + \left( m - 1 \right)d}{b + \left( m - 1 \right) d^1} = \frac{4m - 2 + 3}{6m - 3 + 4}\]
\[ \Rightarrow \frac{a_m}{b_m} = \frac{4m + 1}{6m + 1}\]
संबंधित प्रश्न
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.
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)
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
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.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
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 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
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 are there between 1 and 1000 which when divided by 7 leave remainder 4?
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 :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
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 a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 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 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 arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
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?
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 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. be 3 n2 − n and its common difference is 6, then its first term is
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
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 b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
