The sum of the first three terms of an AP us 18. If the product of the first and third term is 5 times the common difference, find the three numbers.
Answers
Answer:
a=2&15 and d=4&-9
Step-by-step explanation:
place this values and find a=d &a=2d
Answer:
2, 6 , 10
Step-by-step explanation:
Let the common difference be d and the first term be a.
The second term = a + d
Third term = a + 2d
Analyzing the information in the question we have :
The formula for getting the sum of an ap is given by:
Sₙ = n/2(2a + (n - 1)d)
a = first term
n = Number of terms
d = common difference
Doing the substitution we have :
S₃ = 3/2(2a + (3 - 1)d)
18 = 3/2(2a + 2d)
36 = 3(2a + 2d)
6a + 6d = 36....1)
The product of the first and the third term:
a(a + 2d) = 5d
a² + 2ad = 5d.....2)
Dividing through 1 by 6 we have :
a + d = 6
a = 6 - d
Substituting this in 2 we have :
(6 - d)² + 2(6 - d)d = 5d
36 - 12d + d² + 12d - 2d² = 5d
-d² - 5d + 36 = 0
Dividing through by -1 we have :
d² + 5d - 36 = 0
The roots are :
9 and -4
We expand the quadratic equation as follows:
d² + 9d - 4d - 36 = 0
d(d + 9) - 4(d + 9) = 0
(d + 9)(d - 4) = 0
d = -9 or 4
We take 4:
a = 6 - 4 = 2
The terms are as follows:
first term = 2
Second term = 2 + 4 = 6
Third term = 2 + 8 = 10
If we took -9 as the common difference:
a = 6 -(-9) = 15
Second term = 15 - 9 = 6
Third term = 15 - 18 = -3
We therefore take the common difference as 4 to avoid negative integers.
The numbers are thus :