Advertisements
Advertisements
प्रश्न
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
उत्तर १
Let the sum of n terms of the given A.P. be 116.
उत्तर २
In the given problem, we have the sum of the certain number of terms of an A.P. and we need to find the last term for that A.P.
So here, let us first find the number of terms whose sum is 116. For that, we will use the formula,
`S_n = n/2[2a+ (n-1)d]`
Where; a = first term for the given A.P.
d = common difference of the given A.P.
n = number of terms
So for the given A.P (25, 22, 19, .....)
The first term (a) = 25
The sum of n terms `S_n = 116`
Common difference of the A.P (d) = `a_2 - a_1`
= 22 - 25
= -3
So, on substituting the values in the formula for the sum of n terms of A.P. we get
`116 = n/2 [2(25) + (n - 1)(-3)]`
`116 = (n/2)[50 + (-3n + 3)]`
`116 = (n/2) [53 - 3n]`
`(116)(2) = 53n - 3n^2`
So, we get the following quadratic equation,
`3n^2 - 53n + 232 = 0`
On solving by splitting the middle term,we get
`3n^2 - 24n - 29n + 232 = 0`
3n(n - 8) - 29(n - 8) = 0
(3n - 29)(n - 8) = 0
Further
3n - 29 = 0
`n = 29/3`
Also
n - 8 = 0
Now since n cannot be a fraction, so the number of terms is a8
`a_8 = a_1 + 7d`
= 25 + 7(-3)
= 25 - 21
= 4
Therefore the last term of the given A.P such that the sum of terms is 116 is 4
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
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.
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the A.P. 4, 9, 14, ... is 254?
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 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 the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
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.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, 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 farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
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
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 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)`
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
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).
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
