the first term of an ap is 3 and the eleventh term is 18 find the number of term in the progression if the sum is 81
10d = 15
n/2 (2*3+(n-1)*3/2) = 81
To find the number of terms in the arithmetic progression (AP), we need to determine the common difference first.
The formula to find the nth term of an AP is given by:
an = a1 + (n - 1)d
Where:
an = nth term of the AP
a1 = first term of the AP
d = common difference
n = number of terms
Using the given information, we know that:
a1 = 3
a11 = 18
To find the common difference (d):
a11 = a1 + (11 - 1)d
18 = 3 + 10d
15 = 10d
d = 15/10
d = 1.5
Now, we can use the sum formula for an AP to find the number of terms:
Sn = (n/2) * (a1 + an)
Given that the sum (Sn) is 81, we can write the equation as:
81 = (n/2) * (3 + a1 + (n - 1)d)
81 = (n/2) * (3 + 3 + (n - 1) * 1.5)
81 = (n/2) * (6 + 1.5n - 1.5)
81 = (n/2) * (4.5 + 1.5n)
Multiplying through by 2 to eliminate the fraction:
162 = n * (4.5 + 1.5n)
162 = 4.5n + 1.5n^2
162 = 1.5n^2 + 4.5n
0 = 1.5n^2 + 4.5n - 162
Simplifying the quadratic equation, we get:
n^2 + 3n - 108 = 0
Factoring the equation:
(n - 9)(n + 12) = 0
Setting each factor equal to zero:
n - 9 = 0 or n + 12 = 0
Solving for n:
n = 9 or n = -12
Since the number of terms cannot be negative, we disregard n = -12.
Therefore, the number of terms in the arithmetic progression is 9.
To find the number of terms in the arithmetic progression (AP), we can use the formula for the nth term of an AP and the formula for the sum of an AP.
The formula for the nth term of an AP is:
an = a + (n - 1)d
where:
an = nth term
a = first term
n = number of terms
d = common difference
Given that the first term (a) is 3 and the eleventh term (a11) is 18, we can substitute these values into the formula:
a11 = a + (11 - 1)d
18 = 3 + 10d
Simplifying the equation, we have:
18 - 3 = 10d
15 = 10d
Dividing both sides by 10, we get:
d = 1.5
Now we can use the formula for the sum of an AP to find the number of terms.
The formula for the sum of an AP is:
Sn = (n / 2)(2a + (n - 1)d)
Given that the sum (Sn) is 81, we can substitute the values into the formula and solve for n:
81 = (n / 2)(2(3) + (n - 1)(1.5))
Simplifying the equation, we have:
81 = (n / 2)(6 + 1.5n - 1.5)
Multiplying both sides by 2 to eliminate the fraction, we get:
162 = n(7.5 + 1.5n - 1.5)
Expanding and rearranging the terms, we have:
162 = 7.5n + 1.5n^2 - 1.5n
Combining like terms, we have a quadratic equation:
1.5n^2 + 6n - 162 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Using the quadratic formula, with a = 1.5, b = 6, and c = -162, we have:
n = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values and simplifying, we have:
n = (-6 ± √(6^2 - 4(1.5)(-162))) / (2(1.5))
n = (-6 ± √(36 + 972)) / 3
n = (-6 ± √1008) / 3
To find the number of terms, we need to select the positive solution since n represents the number of terms in the AP. From the positive solution, we can calculate the number of terms:
n = (-6 + √1008) / 3
n ≈ (-6 + 31.8) / 3
n ≈ 25.8 / 3
n ≈ 8.6
Therefore, the number of terms in the AP is approximately 8.6. Since the number of terms in an AP must be a whole number, we can conclude that there are 9 terms in the progression.