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)

a₄ = 14