In an AP, 6th term is half the 4th term and the 3rd term is 15.How many terms are needed to give a sum that is equal to 66?
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Hey there!
Given,
6th Term of an A. P is half the fourth term.
3rd term of A. P is 15 .
Let the first term of this Arithmetic progression be a , common difference be d.
We know that, n th term = a + ( n - 1 ) d.
So,
6th term = a + 5d
4th term = a + 3d ,
According to the question,
2( a + 5d ) = a + 3d
2a + 10d = a + 3d
a = -7d
Also,
Third term = 15
a + 2d = 15
-7d + 2d = 15
-5d = 15
d = -3 .
Now, a = -7( -3) = 21 .
Let x terms of the A. P gives the sum 66 .
Sum of x terms = x/2 [2 a + ( x - 1)d ]
66 = x/2 ( 2*21 + ( x - 1 ) ( -3 )
132 = x ( 42 -3x + 3 )
132 = 3x ( 14 - x + 1 )
44 = x ( - x + 15 )
44 = -x² + 15x
x² - 15x + 44 = 0
x² - 11x -4x + 44 = 0
x( x - 11 ) - 4 ( x -11 ) = 0
( x - 4 ) ( x - 11 ) = 0
Either x = 4 or x = 11 .
Therefore, Either 4 or 11 terms are required to get a sum 66 .
Given,
6th Term of an A. P is half the fourth term.
3rd term of A. P is 15 .
Let the first term of this Arithmetic progression be a , common difference be d.
We know that, n th term = a + ( n - 1 ) d.
So,
6th term = a + 5d
4th term = a + 3d ,
According to the question,
2( a + 5d ) = a + 3d
2a + 10d = a + 3d
a = -7d
Also,
Third term = 15
a + 2d = 15
-7d + 2d = 15
-5d = 15
d = -3 .
Now, a = -7( -3) = 21 .
Let x terms of the A. P gives the sum 66 .
Sum of x terms = x/2 [2 a + ( x - 1)d ]
66 = x/2 ( 2*21 + ( x - 1 ) ( -3 )
132 = x ( 42 -3x + 3 )
132 = 3x ( 14 - x + 1 )
44 = x ( - x + 15 )
44 = -x² + 15x
x² - 15x + 44 = 0
x² - 11x -4x + 44 = 0
x( x - 11 ) - 4 ( x -11 ) = 0
( x - 4 ) ( x - 11 ) = 0
Either x = 4 or x = 11 .
Therefore, Either 4 or 11 terms are required to get a sum 66 .
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