The sum of first three terms of an ap is 18. If the product of the first and the third term is 5 times the common difference find the three numbers
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Step-by-step explanation:
Let the three terms be a,b,c
In anA.P terms are in the form a,a+d,a+2d,a+3d,...
Now, a+b+c=18
=>> a+a+d+a+2d = 18
=>> 3a+3d = 18
=>> a+d = 6-----(1)
Therefore 2nd term is 6
a×(a+2d) = 5d
substituting (d=6-a ) from equation (1)
a×[a+2(6-a)] = 5(6-a)
a×[a+12-2a] = 30-5a
a×[12-a] = 30-5a
-a^2+12a=30-5a
a^2-5a-12a+30=0
a^2-17a+30=0
a^2-2a-15a+30=0
a(a-2)-15(a-2) = 0
(a-15)(a-2) = 0
a=15 or a=2
we take a=2
Therefore first term is a=2
Substituting a=2 in equation (1)
a+d=6
=>>2+d =6
=>>d=4
Third term = a+2d
= 2+2(4)
= 2+8
= 10
Hence the three terms a,b,c are 2,6,10
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