The first term of an AP is -12 and the last term is 40 if the sum of the progression is 196, find the number of terms an the common difference
196 / (40 - 12) = n / 2
d = (40 - -12) / (n - 1)
To find the number of terms and the common difference in an arithmetic progression (AP), we can use the sum formula and the formula for the nth term of an AP.
The sum of an arithmetic progression is given by the formula:
S = (n/2)(a + l)
where S is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, given:
a = -12
l = 40
S = 196
Substituting these values into the sum formula:
196 = (n/2)(-12 + 40)
Simplifying:
196 = (n/2)(28)
Dividing both sides by 28:
7 = n/2
Multiplying both sides by 2:
14 = n
Therefore, the number of terms in the arithmetic progression is 14.
Now, to find the common difference (d), we can use the formula for the nth term of an AP:
l = a + (n-1)d
Substituting the given values:
40 = -12 + (14 - 1)d
Simplifying:
40 = -12 + 13d
Adding 12 to both sides:
52 = 13d
Dividing both sides by 13:
4 = d
Therefore, the common difference in the arithmetic progression is 4.
To solve this problem, we need to use the formulas for the sum of an arithmetic progression (AP):
Sum of an AP:
S = (n/2)(a + l)
where:
S is the sum of the AP,
n is the number of terms in the AP,
a is the first term of the AP, and
l is the last term of the AP.
From the problem statement, we know that the first term (a) is -12 and the last term (l) is 40. Additionally, we are given that the sum (S) is 196.
Let's substitute these values into the formula and solve for n:
196 = (n/2)(-12 + 40)
196 = (n/2)(28)
196 = 14n
14n = 196
n = 196/14
n = 14
Therefore, the number of terms in the arithmetic progression is 14.
To find the common difference (d), we can use the formula:
d = (l - a) / (n - 1)
Substituting the known values, we have:
d = (40 - (-12)) / (14 - 1)
d = (40 + 12) / 13
d = 52 / 13
d = 4
Therefore, the common difference of the arithmetic progression is 4.