Question

The third term of an arithmetic sequence is 7 And the fifth term is 13. Determine the nᵗʰ term of the sequence

Answer 1

The nᵗʰ term of the arithmetic sequence is found using the formula a + (n - 1)d, which yields the nᵗʰ term as 3n - 2, based on the provided third and fifth term values.

Step-by-step explanation:

The student is asking to determine the nᵗʰ term of an arithmetic sequence given that the third term is 7 and the fifth term is 13. To find the nᵗʰ term, we need to establish the common difference and the first term of the sequence.

From the third and fifth term values provided, we can calculate the common difference (d) as follows: 13 (fifth term) - 7 (third term) = 6, and since two terms are in between, we divide by 2 to find the common difference for one term.

Therefore, d = 6 / 2 = 3. Now, we can find the first term (a₁) by subtracting 2 times the common difference from the third term: a₁ = 7 - (2 × 3) = 1. The formula for the nᵗʰ term of an arithmetic sequence is a + (n - 1)d. Substituting the known values gives us:

aₙ = 1 + (n - 1)(3) = 3n - 2

Therefore, the nᵗʰ term of the sequence is 3n - 2, where n represents the term position in the sequence.

Answer 2

`a_(n)` = 3n - 2

Step-by-step explanation:

the nth term of an arithmetic sequence is

`a_(n)` = a₁ + (n - 1)d

a₁ is the first term , d the common difference, n the term number

We have to find a₁ and d

given a₃ = 7 and a₅ = 13 , then

a₁ + 2d = 7 → (1)

a₁ + 4d = 13 → (2)

subtract (1) from (2) term by term to eliminate a₁

(a₁ - a₁ ) + (4d - 2d) = 13 - 7

0 + 2d = 6

2d = 6 ( divide both sides by 2 )

d = 3

substitute d = 3 into either of the 2 equations and solve for a₁

substituting into (1)

a₁ + 2(3) = 7

a₁ + 6 = 7 ( subtract 6 from both sides )

a₁ = 1

the nth term is then

`a_(n)` = 1 + 3(n - 1) = 1 + 3n - 3 = 3n - 2