Advertisements
Advertisements
Question
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
Solution
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).
\[\Rightarrow \frac{2a + \left( 2m - 2 \right)d}{2b + \left( 2m - 2 \right) d^1} = \frac{2\left( 2m - 1 \right) + 5}{3\left( 2m - 1 \right) + 4}\]
\[ \Rightarrow \frac{a + \left( m - 1 \right)d}{b + \left( m - 1 \right) d^1} = \frac{4m - 2 + 3}{6m - 3 + 4}\]
\[ \Rightarrow \frac{a_m}{b_m} = \frac{4m + 1}{6m + 1}\]
APPEARS IN
RELATED QUESTIONS
Find the sum to n terms of the A.P., whose kth term is 5k + 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.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
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 two digit numbers which when divided by 4, yields 1 as remainder.
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.
A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
Find:
10th term of the A.P. 1, 4, 7, 10, ...
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
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:
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 four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of first n natural numbers.
Find the sum of all odd numbers between 100 and 200.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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.
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.
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 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.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
Write the sum of first n odd natural numbers.
Write the sum of first n even natural numbers.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
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
If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
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 n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
