find the common difference of an ap whose first term is 5 abd the sum of its first four terms is half the sum of the next four terms.
Answers
Answer:
Required numeric value of the common difference between the terms is 2.
Step-by-step-explanation:
It is given that the first term of the required arithmetic progression is 5.
From the properties of arithmetic progressions :
- nth term = a + ( n - 1 )d, where a is the first term, n is the number of terms and d is the common difference between the terms of that AP.
Now,
Let the first term of this AP be 5 and common difference between the terms be d.
According to the question :
= > sum of first four terms = 1 / 2 of sum of next four terms.
= > [ 1st term + 2nd term + 3rd term + 4th term ] = 1 / 2 x [ 5th term + 6th term + 7th term + 8th term ]
= > [ a + { a + ( 2 - 1 )d } + { a + ( 3 - 1 )d } + { a + ( 4 - 1 )d } ] = 1 / 2 x [ { a + ( 5 - 1 )d + { a + ( 6 - 1 )d } + { a + ( 7 - 1 )d } + { a + ( 8 - 1 )d } ]
= > [ a + ( a + d ) + ( a + 2d ) + ( a + 3d ) ] = 1 / 2 x [ ( a + 4d ) + ( a + 5d ) + ( a + 6d ) + ( a + 7d ) ]
= > [ a + a + a + a + d + 2d + 3d ] = 1 / 2 x [ a + a + a + a + 4d + 5d + 6d + 7d ]
= > [ 4a + 6d ] = 1 / 2 x [ 4a + 22d ]
= > [ 4a + 6d ] = 1 / 2 x 2( 2a + 11d )
= > 4a + 6d = 2a + 11d
= > 4a - 2a = 11d - 6d
= > 2a = 5d
Since the numeric value of first term of this AP is 5, value of a must be 5.
= > 2( 5 ) = 5d
= > 2 = d
Hence the required numeric value of the common difference between the terms is 2.
Answer:
Step-by-step explanation:
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