Find The Value Of K And The Next Term K-2,5k+12,6-K Will From An Arithmetic Sequence
Find the value of k and the next term k-2,5k+12,6-k will from an arithmetic sequence
Answer:
The value of k = -2.
The next term is a₄=-7k or a₄ = 14.
Step-by-step explanation:
Given:
First term, a₁ = k-2
Second term, a₂ = 5k + 12
Third term, a₃ = 6 - k
Find the value of k using the formula for the common difference, d:
a₂ - a₁ = a₃ - a₂
(5k + 12) - (k - 2) = (6 - k) - (5k + 12)
5k + 12 - k + 2 = 6 - k - 5k - 12
4k + 14 = - 6k - 6
(4k + 14 ⇒ common difference expression between a₂ and a₁)
(-6k - 6 ⇒ common difference expression between a₃ and a₂)
Combine the like terms:
4k + 14 = - 6k - 6
6k + 4k = -6 - 14
10k/10 = -20/10
k = -2
The value of k is -2.
Substitute, k = -2:
a₁ = k-2
a₁ = (-2)-2
a₁ = -4
a₂ = 5k + 12
a₂ = 5(-2) + 12
a₂ = -10 + 12
a₂ = 2
a₃ = 6 - k
a₃ = 6 - (-2)
a₃ = 6 + 2
a₃ = 8
The arithmetic sequence when k = -2:
-4, 2, 8
The common difference, d= 6.
To determine the common difference, d:
a₂ - a₁ = a₃ - a₂
2 - (-4) = 8 - 2
2+4 = 6
6 = 6
d = 6
Find the next term, a₄/fourth term:
As expression:
a₄ = a₃ + d (common difference between a₃ and a₂
a₄ = 6 - k + (-6k - 6)
a₄ = 6 - k - 6k - 6
a₄ = -7k
Substitute k = -2:
a₄ = -7(-2)
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