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
Advertisements
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
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
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.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
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?
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 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:
nth term of the A.P. 13, 8, 3, −2, ...
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all even integers between 101 and 999.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
Find the sum of odd integers from 1 to 2001.
In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.
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).
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.
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?
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?
Write the common difference of an A.P. the sum of whose first n terms is
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
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
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 in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
