Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let a be the first term and d be the common difference. Then,
\[a_4 + a_8 = 24\]
\[ \Rightarrow a + \left( 4 - 1 \right)d + a + \left( 8 - 1 \right)d = 24\]
\[ \Rightarrow a + 3d + a + 7d = 24\]
\[ \Rightarrow 2a + 10d = 24 \]
\[ \Rightarrow a + 5d = 12 . . . (i)\]
\[\text { Also }, a_6 + a_{10} = 34\]
\[ \Rightarrow a + \left( 6 - 1 \right)d + a + \left( 10 - 1 \right)d = 34\]
\[ \Rightarrow a + 5d + a + 9d = 34\]
\[ \Rightarrow 2a + 14d = 34\]
\[ \Rightarrow a + 7d = 17 . . . (ii)\]
\[\text { Solving (i) and (ii), we get }: \]
\[2d = 5\]
\[ \Rightarrow d = \frac{5}{2}\]
\[\text { Substituing the value in (i), we get }: \]
\[a + 5\left( \frac{5}{2} \right) = 12\]
\[ \Rightarrow a + \frac{25}{2} = 12\]
\[ \Rightarrow a = \frac{- 1}{2}\]
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
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 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.
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?
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
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.
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 sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
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, 8, 13, ..., 253
How many numbers of two digit are divisible by 3?
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
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 :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of first n odd natural numbers.
Find the sum of all odd numbers between 100 and 200.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
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 a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
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 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.
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 log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
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 n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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
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
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
