Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text { Let the first term, the common difference and the sum of the first n terms of the first A . P . be} a_1 , d_1 and S_1 , respectively, and those of the second A . P . be a_2 , d_2 and S_2 , respectively . \]
\[\text { Then, we have }, \]
\[ S_1 = \frac{n}{2}\left[ 2 a_1 + \left( n - 1 \right) d_1 \right] \]
\[\text { And, } S_2 = \frac{n}{2}\left[ 2 a_2 + \left( n - 1 \right) d_2 \right]\]
\[\text { Given: } \]
\[ \frac{S_1}{S_2} = \frac{\frac{n}{2}\left[ 2 a_1 + \left( n - 1 \right) d_1 \right]}{\frac{n}{2}\left[ 2 a_2 + \left( n - 1 \right) d_2 \right]} = \frac{7n + 2}{n + 4}\]
\[ \Rightarrow \frac{S_1}{S_2} = \frac{\left[ 2 a_1 + \left( n - 1 \right) d_1 \right]}{\left[ 2 a_2 + \left( n - 1 \right) d_2 \right]} = \frac{7n + 2}{n + 4}\]
\[\text { To find the ratio of the 5th terms of the two A . P . s, we replace n by } (2 \times 5 - 1 = 9)\text { in the above equation }: \]
\[ \Rightarrow \frac{\left[ 2 a_1 + \left( 9 - 1 \right) d_1 \right]}{\left[ 2 a_2 + \left( 9 - 1 \right) d_2 \right]} = \frac{7 \times 9 + 2}{9 + 4}\]
\[ \Rightarrow \frac{\left[ 2 a_1 + \left( 8 \right) d_1 \right]}{\left[ 2 a_2 + \left( 8 \right) d_2 \right]} = \frac{7 \times 9 + 2}{9 + 4} = \frac{65}{13} \]
\[ \Rightarrow \frac{\left[ a_1 + 4 d_1 \right]}{\left[ a_2 + 4 d_2 \right]} = \frac{5}{1} = 5: 1\]
APPEARS IN
संबंधित प्रश्न
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.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
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.
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
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.
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.
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
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 :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of first n natural numbers.
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
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 a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
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. 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:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
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.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
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 ______.
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 ______.
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 ______.
