d = (T11-T1)/10 = 3/2
n/2 (2*3+(n-1)(3/2)) = 81
n = 9
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.
2 answers
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
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