The sum of first three numbers in AP is 18, if the product of the first and third term is 5 times the common difference find three numbers
Answers
a+(a+d)+(a+2d)=18
3a+3d=18
3(a+d)=18
(a+d)=6.........Eq 1
a*(a+2d)=5d
a*(a+d+d) =5d
a*(6+d)=5d
a*(6+6-a)=5(6-a)
a*(12-a)=30-5a
12a-a*a=30-5a
a*a-17a+30=0 [Factorising]
a=15 or a=2...substitute in Eq1
If a=15, d= - 9, not Possible
If a=2, d=4, possible
Then three numbers which are in AP and......
a=2
(a+d)=(2+4)=6
(a+2d)=(2+2*4)=10
The numbers 2,6,10 are in AP
The first three terms of the Arithmetic progression is (2, 6, 10) and (15, 6, -3) when a=(6) and d= (4 and -9)
Step-by-step explanation:
Given Data
(a-d) + a + (a+d) = 18 --------> (1)
(a-d) × (a+d) = 5d ----------> (2)
To find the terms (a-d), a and (a+d)
(2) => (a-d) × (a+d) = 5d
a²-d² = 5d --------> (3)
(1) => (a-d) + a + (a+d) = 18
a-d + a + a+d = 18
3a = 18
a = 6 ------> (4)
Substitute the value of 'a' in (3)
(6)² - d² = 5d
36 - d² = 5d
d² + 5d - 36 = 0
d² + 9d -4d- 36 = 0
d(d+9) - 4(d+9) = 0
d= 4 ; d= -9
a= 6
If d= 4 and a= 6, then Arithmetic progression series is 2, 6, 10
If a= 6 and d= -9, then Arithmetic progression series is 15, 6, -3
Therefore the first three terms of the Arithmetic progression is (2, 6, 10) and (15, 6, -3)
To Learn More ...
1)Sum of three consecutive terms of an Arithmetic Progression is 42 and their product is 2520. Find the terms of the Arithmetic Progression.
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2) In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 find the common difference and the first term
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