Advertisements
Advertisements
Question
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
Advertisements
Solution
\[\text { Let there be two A . P . s } . \]
\[\text { Let their first terms be } a_1 \text { and }a_2 \text { and their common differences be } d_1 \text { and } d_2 . \]
\[\text { Given }: \]
\[ \frac{5n + 4}{9n + 6} = \frac{\text { Sum of n terms in the first A . P } .}{\text { Sum of n terms in the second A . P } .}\]
\[ \Rightarrow \frac{5n + 4}{9n + 6} = \frac{2 a_1 + [(n - 1) d_1 ]}{2 a_2 + [(n - 1) d_2 ]}\]
\[\text { Putting n } = 2 \times 18 - 1 = 35 \text { in the above equation, we get }: \]
\[ \frac{5 \times 35 + 4}{9 \times 35 + 6} = \frac{2 a_1 + 34 d_1}{2 a_2 + 34 d_2}\]
\[ \Rightarrow \frac{179}{321} = \frac{a_1 + 17 d_1}{a_1 + 17 d_1}\]
\[ \Rightarrow \frac{179}{321} = \frac{\text { 18th term of the first A . P } .}{\text { 18th term of the second A . P } .}\]
APPEARS IN
RELATED QUESTIONS
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
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.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
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, ...?
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.
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
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 :
a + b, a − b, a − 3b, ... to 22 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 even integers between 101 and 999.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
If a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
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.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Write the common difference of an A.P. whose nth term is xn + y.
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
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 sum of first n even natural numbers.
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
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms 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
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
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. What is his total earnings during the first year?
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series 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 ______.
