the sum of n terms of an ap is 300. if the first term is 5 and the nth term is 55, how many terms has the ap?
5+(n-1)d = 55
n/2 (2*5 + (n-1)d) = 300
substituting,
n/2 (5+55) = 330
n = 11
To find the number of terms in an Arithmetic Progression (AP), we can use the formula for the nth term of an AP:
an = a1 + (n-1)d
Where:
an = nth term
a1 = first term
d = common difference
We are given that the first term (a1) is 5 and the nth term (an) is 55. Let's substitute these values into the formula:
55 = 5 + (n-1)d
Simplifying the equation:
50 = (n-1)d
Now, we need more information to determine the common difference (d).
To find the number of terms in an arithmetic progression (AP), we need to use the formula for the sum of the first n terms of an AP: Sn = (n/2) × (2a + (n-1)d),
where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Given:
Sn = 300
a = 5
n = ?
the nth term = 55
We know that the sum of the first n terms, Sn, is equal to 300. So we can write the equation as:
300 = (n/2) × (2a + (n-1)d)
Substituting the given values into the equation, we get:
300 = (n/2) × (2(5) + (n-1)d)
Simplifying further:
300 = (n/2) × (10 + (n-1)d)
300 = (n/2) × (10 + nd - d)
Multiplying through by 2 to remove the fraction:
600 = n × (10 + nd - d)
600 = 10n + n²d - nd
Rearranging the equation:
n²d - (d-10)n + 600 = 0
This is a quadratic equation, and to determine the number of terms, we need to solve for n. However, we need an additional piece of information, specifically the value of the common difference (d), to find the number of terms.