The first and the last term of an arithmetic progression are 79 and -5 and it sum is 814 . Find the number of term in an A.P
Let's denote the number of terms in the arithmetic progression as n.
The formula for the sum of an arithmetic progression is given by Sn = n/2(a + l), where Sn is the sum of the progression, a is the first term, l is the last term, and n is the number of terms.
Plugging in the given values, we have:
814 = n/2(79 - 5)
814 = n/2(74)
814 = 37n
Dividing both sides by 37, we get:
n = 22
Therefore, the number of terms in the arithmetic progression is 22.
Let's assume that the number of terms in the arithmetic progression is 'n'.
The first term of the arithmetic progression is 79.
The last term of the arithmetic progression is -5.
The sum of an arithmetic progression is given by the formula:
S = (n/2) * (first term + last term)
Substituting the given values into the formula, we have:
814 = (n/2) * (79 + (-5))
Simplifying further:
814 = (n/2) * 74
To eliminate the fraction, we can multiply both sides of the equation by 2:
1628 = n * 74
Now, we can divide both sides of the equation by 74 to isolate 'n':
n = 1628 / 74
Simplifying:
n ≈ 22
Therefore, the number of terms in the arithmetic progression is approximately 22.
To find the number of terms in an arithmetic progression (A.P.), we need to use the following formula:
Sum of A.P. = [n/2] * [2a + (n-1)d]
Where:
- n is the number of terms in the A.P.
- a is the first term of the A.P.
- d is the common difference between terms.
In this problem, we are given:
- The first term (a) is 79.
- The last term (which is also the n-th term) is -5.
To find the common difference (d), we can use the formula for the n-th term of an A.P.:
n-th term = a + (n-1)d
Substituting the given values:
-5 = 79 + (n-1)d
Now, let's find the common difference (d):
-5 = 79 + nd - d
-5 - 79 = nd - d
-84 = (n-1)d
Next, we use the sum of A.P. formula with the given values:
814 = [n/2] * [2(79) + (n-1)d]
Substituting the value of (n-1)d from the previous equation:
814 = [n/2] * [158 - 84]
Simplifying:
814 = [n/2] * 74
Now, to solve for n, we'll isolate n by multiplying both sides of the equation by 2/74:
814 * 2/74 = n
n = 814 * 2 / 74
Calculating this:
n ≈ 22.05
Since the number of terms (n) must be a whole number, we round down to the nearest integer:
n = 22.
Therefore, the number of terms in the arithmetic progression is 22.