The first term of an AP is 3 and the eleventh term is 18.find the number of terms in the progression if the sum is 81.
d = (T11-T1)/10 = 3/2
n/2 (2*3+(n-1)(3/2)) = 81
n = 9
n-th member in AP
an = a1 + ( n - 1 ) d
d = the difference between successive terms
In this case :
a11 = a1 + ( 11 - 1 ) d
a11 = a1 + 10 d = 18
18 = 3 + 10 d Subtract 3 to both sides
18 - 3 = 3 + 10 d - 3
15 = 10 d Divide both sides by 10
15 / 10 = 10 d / 10
1.5 = d
d = 1.5
The sum of the n terms of an arithmetic progression:
Sn = n [ 2 a1 + ( n - 1 ) d ] / 2
In this case
a1 = 3
d = 1.5
Sn = 81
so :
81 = n * [ 2 * 3 + ( n - 1 ) * 1.5 ] / 2
81 = n * [ 6 + 1.5 n - 1.5 ] / 2
162 = n * [ 6 + ( n - 1 ) * 1.5 ]
81 = n * ( 4.5 + 1.5 n ) / 2 Multiply both sides by 2
162 = n * ( 4.5 + 1.5 n )
162 = 4.5 n + 1.5 n ^ 2 Subtract 162 to both sides
162 - 162 = 4.5 n + 1.5 n ^ 2 -162
0 = 4.5 n + 1.5 n ^ 2 -162
1.5 n ^ 2+ 4.5 n -162 = 0
Solutions :
n = - 12
and
n = 9
Number of members can't be negative number so n = 9
Your AP :
3, 4.5, 6, 7.5 ,9, 10.5, 12, 13.5, 15
To find the number of terms in the arithmetic progression (AP), we need to use the following formula for the nth term:
an = a1 + (n - 1)d
where:
an = nth term
a1 = first term
n = number of terms
d = common difference
Given that the first term is 3 (a1 = 3) and the eleventh term is 18, we can find the common difference (d) using the formula:
a11 = a1 + (11 - 1)d
18 = 3 + 10d
10d = 15
d = 1.5
Now, we can find the sum of the arithmetic progression using the formula:
Sn = (n/2)(a1 + an)
Substituting the given values:
81 = (n/2)(3 + 3 + (n - 1)(1.5))
81 = (n/2)(6 + 1.5n - 1.5)
81 = (n/2)(4.5 + 1.5n)
162 = n(4.5 + 1.5n)
162 = 4.5n + 1.5n^2
1.5n^2 + 4.5n - 162 = 0
Simplifying the equation, we get a quadratic equation:
n^2 + 3n - 108 = 0
Factoring or using the quadratic formula, we find that:
(n - 9)(n + 12) = 0
Therefore, the possible solutions are:
n - 9 = 0 or n + 12 = 0
If n - 9 = 0, then n = 9
If n + 12 = 0, then n = -12 (but since the number of terms cannot be negative, we disregard this solution)
Hence, the number of terms in the arithmetic progression is 9.
To find the number of terms in an arithmetic progression (AP), we can use the formula for the nth term of an AP:
nth term = a + (n - 1)d,
where "a" is the first term, "n" is the number of terms, and "d" is the common difference.
Given that the first term (a) is 3 and the eleventh term is 18, we can substitute these values into the formula to find the common difference (d):
18 = 3 + (11-1)d
18 = 3 + 10d
15 = 10d
d = 1.5
Now that we know the common difference (d), we can find the sum of an AP using the formula:
Sum of n terms (Sn) = (n/2)(2a + (n-1)d),
where Sn is the sum of n terms.
Given that the sum (Sn) is 81, we can substitute the known values into the formula:
81 = (n/2)(2*3 + (n-1)*1.5)
81 = (n/2)(6 + 1.5n - 1.5)
81 = (n/2)(4.5 + 1.5n)
81 = 4.5n + 1.5n^2/2 - 1.5n/2
Now, we have a quadratic equation. Simplifying this equation, we get:
81 = 4.5n + 0.75n^2 - 0.75n
0 = 0.75n^2 + 3.75n - 81
Now, we can solve this quadratic equation to find the value of "n". We can use factoring, completing the square, or the quadratic formula:
Applying the quadratic formula, the equation becomes:
n = (-b ± sqrt(b^2 - 4ac)) / (2a)
Where a = 0.75, b = 3.75, and c = -81. Plugging in these values:
n = (-3.75 ± sqrt(3.75^2 - 4*0.75*(-81))) / (2*0.75)
Now, using a calculator to simplify the expression, we get:
n ≈ 8.065 or n ≈ -13.398
Since the number of terms cannot be negative, the approximate number of terms in the arithmetic progression is 8. Therefore, there are 8 terms in the progression.