Advertisements
Advertisements
Question
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Advertisements
Solution
\[\text { Let the four numbers be }a - 3d, a - d, a + d, a + 3d . \]
\[\text { Sum } = 50\]
\[ \Rightarrow a - 3d + a - d + a + d + a + 3d = 50\]
\[ \Rightarrow 4a = 50\]
\[ \Rightarrow a = \frac{25}{2} . . . (i)\]
\[\text { Also }, a + 3d = 4(a - 3d)\]
\[ \Rightarrow a + 3d = 4a - 12d\]
\[ \Rightarrow 3a = 15d\]
\[ \Rightarrow a = 5d\]
\[ \Rightarrow \frac{25}{2 \times 5} = d \left( \text { Using } (i) \right)\]
\[ \Rightarrow \frac{5}{2} = d\]
\[\text { So, the terms are as follows: } \]
\[ \left( \frac{25}{2} - 3 \times \frac{5}{2} \right), \left( \frac{25}{2} - \frac{5}{2} \right), \left( \frac{25}{2} + \frac{5}{2} \right), \left( \frac{25}{2} + 3 \times \frac{5}{2} \right)\]
\[ = 5, 10, 15, 20\]
RELATED QUESTIONS
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
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, ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
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.
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all integers between 50 and 500 which are divisible by 7.
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
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
If a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
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.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
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.
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
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
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
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
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
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?
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
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 ______.
Which of the following best defines an Arithmetic Progression (A.P.)?
Which type of A.P. is represented by the sequence \(2, 5, 8, 11, \ldots\)?
Which of the following is an example of a Constant A.P.?
