The first and the last term of ap is 21 and-47 ,if the sum of series is 234 calculate ,number of term ,common difference, thesum of first 18 terms
S = n/2 (2a+d)
Plugging your numbers, I don't get a positive integer ...
my bad
S = n/2 (a1+an) = n/2 (21 + 47) = 234
Hmm. Still not a positive integer for n.
To find the number of terms, the common difference, and the sum of the first 18 terms of an arithmetic progression (AP), we need to use the given information.
We are given:
The first term, a₁ = 21
The last term, aₙ = -47
The sum of the series, Sₙ = 234
First, let's find the number of terms (n):
The formula to find the nth term of an AP is:
aₙ = a₁ + (n - 1) * d
We know aₙ = -47 and a₁ = 21, so we can substitute these values into the formula:
-47 = 21 + (n - 1) * d
Next, we'll use the formula to find the sum of the series (Sₙ):
The formula for the sum of the first n terms of an AP is:
Sₙ = (n / 2) * (a₁ + aₙ)
Substituting the given values into the formula:
234 = (n / 2) * (21 + (-47))
Now, we have two equations with two variables (n and d):
-47 = 21 + (n - 1) * d
234 = (n / 2) * (-26)
To solve these equations simultaneously, we'll use substitution.
From the first equation, we can express d in terms of n:
(n - 1) * d = -47 - 21
(n - 1) * d = -68
d = -68 / (n - 1)
Now, we substitute this value of d into the second equation:
234 = (n / 2) * (-26)
234 = -13n
n = -234 / 13
n ≈ 18
So, the number of terms (n) is approximately 18.
To find the common difference (d), substitute n = 18 into the first equation:
-47 = 21 + (18 - 1) * d
-47 = 21 + 17d
17d = -68
d = -68 / 17
d = -4
So, the common difference (d) is -4.
Finally, to find the sum of the first 18 terms:
S₁₈ = (18 / 2) * (21 - 47)
S₁₈ = 9 * (-26)
S₁₈ = -234
Therefore, the sum of the first 18 terms is -234.