In an arithmetic sequence where a6=13 and a10=25, what is the explicit formula for the general term an?

3n-5

To find the explicit formula for the general term, we need to determine the common difference (d) in the arithmetic sequence.

First, let's find the value of a6. We know that a6 = 13.

We can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n-1)d

where a1 is the first term and d is the common difference.

Substituting a6 = 13 and n = 6 into the formula, we have:

13 = a1 + (6-1)d

13 = a1 + 5d ----- Equation (1)

Next, let's find the value of a10. We know that a10 = 25.

Again, using the formula for the nth term:

an = a1 + (n-1)d

Substituting a10 = 25 and n = 10, we have:

25 = a1 + (10-1)d

25 = a1 + 9d ----- Equation (2)

Now, we have a system of two equations: Equation (1) and Equation (2).

We can solve this system of equations to find the values of a1 (the first term) and d (the common difference).

Subtracting Equation (1) from Equation (2), we have:

25 - 13 = (a1 + 9d) - (a1 + 5d)

12 = 4d

Dividing both sides by 4, we get:

d = 3

Now, we can substitute the value of d into either Equation (1) or Equation (2) to find the value of a1. Let's use Equation (1):

13 = a1 + 5(3)

13 = a1 + 15

Subtracting 15 from both sides, we get:

-2 = a1

So, the first term (a1) is -2 and the common difference (d) is 3.

Therefore, the explicit formula for the general term (an) in this arithmetic sequence is:

an = -2 + (n-1)(3)

or alternatively,

an = -2 + 3n - 3

Simplifying, we have:

an = 3n - 5

Thus, the explicit formula for the general term in this arithmetic sequence is an = 3n - 5.

To find the explicit formula for the general term, we need to determine the common difference (d) of the arithmetic sequence. Once we have the common difference, we can use the formula an = a1 + (n - 1)d to find any term in the sequence.

Given a6 = 13 and a10 = 25, we can deduce the following:

a6 = a1 + 5d (Since the sixth term is a1 + 5d)
a10 = a1 + 9d (Since the tenth term is a1 + 9d)

Subtracting the first equation from the second equation, we get:

a10 - a6 = (a1 + 9d) - (a1 + 5d)
25 - 13 = 4d
12 = 4d
d = 3

Now that we have the common difference (d = 3), we can find the explicit formula for the general term an. It is an = a1 + (n - 1)d.

So, the explicit formula for the general term an in this arithmetic sequence is:

an = a1 + (n - 1)d

where a1 is the first term and d is the common difference.

an = ao + (n-1)d

where
an = the nth term
ao = the first term
n = number of terms
d = difference between two consecutive terms

Substituting the given,
(i) a6 = 13 (here, n=6, an=13)
13 = ao + (6-1)d
13 = ao + 5d

(ii) a10 = 25 (here, n=10, an=25)
25 = ao + (10-1)d
25 = ao + 9d

Now we have two linear equations. We can do elimination method to get d:
13 = ao + 5d
-(25 = ao + 9d)
--------------------------
-12 = -4d
d = 12/4
d = 3

Solving for ao,
13 = ao + 5(3)
13 = ao + 15
ao = 13 - 15
ao = -2

Therefore, the general expression for an:
an = -2 + 3*(n-1)

Hope this helps~ `u`