Advertisements
Advertisements
प्रश्न
Solve the equation
– 4 + (–1) + 2 + ... + x = 437
Advertisements
उत्तर
Given equation is,
– 4 + (–1) + 2 + ... + x = 437 ...(i)
Here, – 4 – 1 + 2 + ... + x forms an AP with first term = – 4,
Common difference = – 1 – (– 4) = 3,
an = l = x
∵ nth term of an AP,
an = l = a + (n – 1)d
`\implies` x = – 4 + (n – 1)3 ...(ii)
`\implies` `(x + 4)/3` = n – 1
`\implies` n = `(x + 7)/3`
∴ Sum of an AP,
Sn = `n/2[2a + (n - 1)d]`
Sn = `(x + 7)/(2 xx 3)[2(-4) + ((x + 4)/3) * 3]`
= `(x + 7)/(2 xx 3)(-8 + x + 4)`
= `((x + 7)(x - 4))/(2 xx 3)`
From equation (i),
Sn = 437
`\implies ((x + 7)(x - 4))/(2 xx 3)` = 437
`\implies` x2 + 7x – 4x – 28 = 874 × 3
`\implies` x2 + 3x – 2650 = 0
x = `(-3 +- sqrt((3)^2 - 4(-2650)))/2` ...[By quadratic formula]
= `(-3 +- sqrt(9 + 10600))/2`
= `(-3 +- sqrt(10609))/2`
= `(-3 +- 103)/2`
= `100/2, (-106)/2`
= 50, – 53
Here, x cannot be negative i.e., x ≠ – 53
Also for x = – 53, n will be negative which is not possible
Hence, the required value of x is 50.
APPEARS IN
संबंधित प्रश्न
The sum of n terms of three arithmetical progression are S1 , S2 and S3 . The first term of each is unity and the common differences are 1, 2 and 3 respectively. Prove that S1 + S3 = 2S2
In an AP given an = 4, d = 2, Sn = −14, find n and a.
Find the sum of the first 15 terms of each of the following sequences having the nth term as
yn = 9 − 5n
The 19th term of an AP is equal to 3 times its 6th term. If its 9th term is 19, find the AP.
Draw a triangle PQR in which QR = 6 cm, PQ = 5 cm and times the corresponding sides of ΔPQR?
Write an A.P. whose first term is a and common difference is d in the following.
Choose the correct alternative answer for the following question .
If for any A.P. d = 5 then t18 – t13 = ....
Simplify `sqrt(50)`
If the seventh term of an A.P. is \[\frac{1}{9}\] and its ninth term is \[\frac{1}{7}\] , find its (63)rd term.
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 − kSn−1 + Sn−2, then k =
If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is
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
Q.17
If the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.
Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.
Find the sum of those integers from 1 to 500 which are multiples of 2 or 5.
[Hint (iii) : These numbers will be : multiples of 2 + multiples of 5 – multiples of 2 as well as of 5]
An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Find the AP.
The sum of first five multiples of 3 is ______.
If Sn denotes the sum of first n terms of an AP, prove that S12 = 3(S8 – S4)
In a ‘Mahila Bachat Gat’, Kavita invested from the first day of month ₹ 20 on first day, ₹ 40 on second day and ₹ 60 on third day. If she saves like this, then what would be her total savings in the month of February 2020?
