math algebraically

Question

. What is the general sequence equation for this situation?

. An arithmetic sequence has a 10th term of 15 and 14th term of 35.

Answer 1

The general formula for an arithmetic sequence is given by a_n = a₁ + (n - 1) d where a₁ is the first term of the sequence and d is the common difference.

Using the formula, we have

a₁₀ = a₁ + (10 -1)d = 15 ⇒ a₁ + 9d = 15 (1)

a₁₄ = a₁ + (14 -1)d = 35 ⇒ a₁ + 13d = 35 (2)

Subtract equation (1) from equation (2) we get

13d - 9d = 35 - 15

4d = 20

d = 5

Now replace d in either equation (1) or (2) to find a₁

a₁+ 45 = 15

a₁ = 15 - 45 = -30

Hence;

a_n = a₁ + (n - 1)d = -30 + (n - 1)(5) = -30 + 5n - 5 = 5n - 35.

a_n = 5n - 35 is the general form of this sequence.

Answer 2

a₁₄ = a₁₀ + 4d

35 = 15 + 4d

20 = 4d

5 = d

a_n+1 = a_n + 5