The first and last terms of an
A.P are and 90, if the sum of the
term is 1, find the number of
terms.
A. 12
B. 15
C. 22
Let's denote the first term of the A.P as 'a' and the common difference as 'd'.
The first term is 'a' and the last term is 90, so we can write the equation:
a + (n-1)d = 90 (1), where 'n' is the number of terms.
The sum of the terms in an A.P is given by the formula:
S = (n/2)(2a + (n-1)d)
Given that the sum of the terms is 1, we can write the equation:
(n/2)(2a + (n-1)d) = 1
Substituting the value of 'a' from equation (1) into the equation (2), we get:
(n/2)(2(90 - (n-1)d) + (n-1)d) = 1
Simplifying the equation, we have:
(n/2)(180 - 2nd + d) = 1
n(90 - nd + d/2) = 1*2
n(90 + (d/2) - nd) = 2
90n + (d/2)n - nd^2 = 2
As we can see, this equation is quadratic in 'n'.
To solve this equation, we need one more equation.
Since the first term is 'a', we can write:
a + (n-1)d = 90
Substituting the values of 'a' and the sum of the terms into this equation, we get:
90 - nd + d = 90
d = nd
Dividing both sides by d, we get:
n = 1
Substituting this value of 'n' into the equation 90n + (d/2)n - nd^2 = 2, we get:
90*1 + (d/2)*1 - d^2 = 2
90 + (d/2) - d^2 = 2
Rearranging the equation and simplifying, we get a quadratic equation:
d^2 - (d/2) + 88 = 0
Using the quadratic formula, we can solve for 'd':
d = (-(-1/2) ± √((-1/2)^2 - 4(1)(88))) / (2(1))
d = (1/2 ± √(1/4 - 352)) / 2
d = (1/2 ± √(-351/4)) / 2
The discriminant is negative, so there are no real solutions for 'd'.
Therefore, it is not possible to determine the number of terms of the A.P in this case.
Let's call the first term of the arithmetic progression (AP) "a" and the common difference "d".
The first term is a = 1, and the last term is l = 90.
The sum of n terms in an arithmetic progression is given by the formula: Sn = (n/2) * (a + l), where Sn represents the sum of n terms.
We know that Sn = 1, so we can substitute the values into the equation:
1 = (n/2) * (1 + 90)
1 = (n/2) * 91
1 = (91n)/2
2 = 91n
n = 2/91
The number of terms, n, must be an integer, so we need to find the smallest positive integer value for n. Let's calculate the value of n:
n = 2/91 = 0.021978
Since n must be a positive integer, the closest integer value for n is 1.
Therefore, the number of terms in the arithmetic progression is 1.
The correct answer is not among the options given.