precal question

Question

Determine the common difference, the fifth term, the nth term, and the 100th term of the arithmetic sequence.

14, 10, 6, 2,   

d	=
a5	=
an	=
a100	=

Answer 1

d = -4 since the numbers are counting down by 4 each time.

a₅ is the next number in the sequence, which would be 2 - 4 or -2.

To find the nth term, know that arithmetic sequences of a linear relationship. In other words, we can write a_n as a linear function.

a_n = mn + b, where m is the slope and b is the y-intercept. The slope is -4 since every time n increases by 1, a_n decreases by 4.

a_n = -4n + b. Now let's find b. We can do that by plugging in a point. Let n = 1

a₁ = -4*1 + b.

a₁ is the first term in the sequence so a₁ = 14

14 = -4 + b. Therefore b = 18, and the equation is:

a_n = -4n +18

Lastly to get a₁₀₀, just plug 100 in for n into the equation

a₁₀₀ = -4(100) + 18 = -400 + 18 = - 382

Answer 2

The common difference, d, is - 4. (Just look to see how much you ad or subtract to each term to get the next.)

For arithmetic sequences, the formula follows: a_n = a₁ + d(n - 1). Where a₁ is the first term (in this case 14).

To get a₅, plug in a₁ = 14, d = - 4, and n = 5: a₅ = 14 - 4(5 - 1). a₅ = - 2. Similarly a₁₀₀ = 14 - 4(100 - 1) = - 382.

a_n = 14 - 4(n - 1). Some teachers and textbooks would “simplify” this to a_n = - 4n + 18.