if the second term of an arithmetic sequence is 15 and the fifth term is 24. determine the third and fourth term of the sequence
a + (n-1) d
second is a + d = 15
fifth is a + 4 d = 24
so a = 15 - d
and
(15-d) + 4 d = 24
3 d = 9
d = 3
then a = 12
so term n is
12 + (n-1)3
To determine the third and fourth terms of the arithmetic sequence, we need the common difference (d) between the terms.
The formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
Here, a1 is the first term and n is the position of the term in the sequence.
Given:
a2 = 15
a5 = 24
First, let's find the common difference (d).
Using the formula, we can substitute the given values:
a2 = a1 + (2 - 1)d
15 = a1 + d ...(1)
a5 = a1 + (5 - 1)d
24 = a1 + 4d ...(2)
Now we have a system of equations (1) and (2) to solve.
We can solve the system either by substitution or elimination. Let's use substitution.
Substitute equation (1) into equation (2) to eliminate the variable a1:
24 = (15 - d) + 4d
24 = 15 + 3d
3d = 24 - 15
3d = 9
d = 3
Now that we know the value of d, we can substitute it back into equation (1) to find a1:
15 = a1 + 3
a1 = 15 - 3
a1 = 12
Now we can use the formula an = a1 + (n - 1)d to find the third and fourth terms:
a3 = 12 + (3 - 1) * 3
a3 = 12 + 2 * 3
a3 = 12 + 6
a3 = 18
a4 = 12 + (4 - 1) * 3
a4 = 12 + 3 * 3
a4 = 12 + 9
a4 = 21
Therefore, the third term is 18, and the fourth term is 21.
To determine the third and fourth terms of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
𝑎_𝑛 = 𝑎_1 + (𝑛−1)𝑑
where 𝑎_𝑛 is the nth term of the sequence, 𝑎_1 is the first term, 𝑛 is the position of the term, and 𝑑 is the common difference between terms.
Given that the second term of the sequence (𝑎_2) is 15, we can substitute 𝑎_2 = 15 and 𝑛 = 2 into the formula to find 𝑑:
15 = 𝑎_1 + (2−1)𝑑
15 = 𝑎_1 + 𝑑
Similarly, given that the fifth term of the sequence (𝑎_5) is 24, we can substitute 𝑎_5 = 24 and 𝑛 = 5 into the formula to find 𝑑:
24 = 𝑎_1 + (5−1)𝑑
24 = 𝑎_1 + 4𝑑
Now we have two equations:
15 = 𝑎_1 + 𝑑
24 = 𝑎_1 + 4𝑑
We can solve this system of equations to find the values of 𝑎_1 and 𝑑:
Subtracting the first equation from the second equation, we get:
24 - 15 = (𝑎_1 + 4𝑑) - (𝑎_1 + 𝑑)
9 = 4𝑑 - 𝑑
9 = 3𝑑
Divide both sides of the equation by 3:
𝑑 = 3
Substituting the value of 𝑑 back into one of the original equations, we can solve for 𝑎_1:
15 = 𝑎_1 + 3
12 = 𝑎_1
So, the first term (𝑎_1) is 12 and the common difference (𝑑) is 3.
Now we can use the formula for the nth term to find the third term (𝑎_3) and fourth term (𝑎_4) of the sequence:
𝑎_3 = 𝑎_1 + (3−1)𝑑
𝑎_3 = 12 + 2(3)
𝑎_3 = 12 + 6
𝑎_3 = 18
𝑎_4 = 𝑎_1 + (4−1)𝑑
𝑎_4 = 12 + 3(3)
𝑎_4 = 12 + 9
𝑎_4 = 21
Therefore, the third term is 18 and the fourth term is 21.