Find the first term and the common difference of the arithmetic sequence if a2-a6+a4=-7 and a8-a7=2a4

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The nth term of an arithmetic sequence can be written as a + (n - 1)*d, where a is the first term and d is the common difference.

We have a2-a6+a4=-7

=> a + d - a - 5d + a + 3d = -7

=> a - d = -7

=> d = a + 7

a8 - a7 = 2*a4

=> a + 7d - a - 6d = 2*(a + 3d)

=> d = 2a + 6d

=> 2a + 5d = 0

Now substitute d = a + 7 in  2a + 5d = 0

=> 2a + 5(a + 7) = 0

=> 2a + 5a + 35 = 0

=> 7a + 35 = 0

=> a = -35/7

=> a = -5

d = a + 7 = -5 + 7 = 2

The first term is -5 and the common difference is 2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll write both given relation from enunciation with respect to the first term and the common difference:

a2-a6+a4 = (a1 + d) - (a1 + 5d) + (a1 + 3d)

We'll eliminate and combine like terms and we'll get:

a1 - d = -7

The first equation has been changed into a1 - d = -7.

We'll write the 2nd relation from enunciation with respect to the first term and the common difference:

a8-a7 = (a1 + 7d) - (a1 + 6d)

We'll eliminate and combine like terms and we'll get:

d = 2a4

d = 2(a1 + 3d)

d = 2a1 + 6d

2a1 + 5d = 0

a1 = -5d/2

-5d/2  - d = -7

-5d - 2d = -14

-7d = -14

d = 2

a1 = -5*2/2

a1 = -5

The first term and the common difference of the arithmetical sequence are: a1 = -5 and d = 2.

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