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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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
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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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