The fifth term of an arithmetic sequence is equal to 6 and the sum of the first 12 terms is equal to 45. Find the first term and the common difference.

Why did the arithmetic sequence go to therapy? It had problems summing up its emotions!

Let's write down what we know. The fifth term of the arithmetic sequence is 6, so we have:

a + 4d = 6 (where a is the first term and d is the common difference)

We also know that the sum of the first 12 terms is equal to 45, so:

12/2 * (2a + (12-1)d) = 45

Simplifying the equation we get:

6(2a + 11d) = 45

Now, solving this equation along with the first equation will help us find both the first term (a) and the common difference (d). However, I must warn you, solving equations is always a little "arithmetic-ing." Let's get to it!

To solve this problem, we need to use the formulas for the nth term and the sum of an arithmetic sequence.

The nth term of an arithmetic sequence is given by the formula:
an = a1 + (n - 1)d

The sum of the first n terms of an arithmetic sequence is given by the formula:
Sn = (n/2)(a1 + an)

We are given that the fifth term (n = 5) is equal to 6:
a5 = a1 + (5 - 1)d = 6

We are also given that the sum of the first 12 terms (n = 12) is equal to 45:
S12 = (12/2)(a1 + a12) = 45

Let's solve these equations simultaneously to find the first term (a1) and the common difference (d).

From the first equation, we have:
a5 = a1 + 4d = 6 ...(1)

From the second equation, we have:
S12 = (12/2)(a1 + a12) = 45
6(a1 + a12) = 45
a1 + a12 = 7.5 ...(2)

Now, let's solve these equations simultaneously:
From Equation (1), we can express a1 in terms of d:
a1 = 6 - 4d

Substituting this value of a1 in Equation (2), we have:
6 - 4d + a12 = 7.5
a12 = 7.5 - 6 + 4d
a12 = 1.5 + 4d

Now, we can use these equations to find the values of a12 and a1 + a12:
a1 + a12 = 7.5
(6 - 4d) + (1.5 + 4d) = 7.5
6 + 1.5 = 7.5
7.5 = 7.5

This shows that our calculations are correct.

Therefore, the first term (a1) is 6 - 4d and the common difference (d) is any value as long as the equation a12 = 1.5 + 4d is satisfied.

To find the first term and the common difference of an arithmetic sequence, we can use the formulas given by:

nth term (an) = a + (n-1)d
sum of n terms (Sn) = (n/2)(2a + (n-1)d)

Let's solve the problem step-by-step:

1. Given that the fifth term (a5) is 6, we can substitute these values into the nth term formula:
6 = a + (5-1)d
6 = a + 4d -- equation (1)

2. We are also given that the sum of the first 12 terms (S12) is 45. Substituting these values into the sum of n terms formula:
45 = (12/2)(2a + (12-1)d)
45 = 6(2a + 11d)
45 = 12a + 66d -- equation (2)

Now, we have a system of equations (equation 1 and equation 2) involving two variables (a and d). Let's solve this system:

Using equation (1), we can express a in terms of d:
a = 6 - 4d -- equation (3)

Substituting equation (3) into equation (2):
45 = 12(6 - 4d) + 66d
45 = 72 - 48d + 66d
45 = 72 + 18d
18d = 45 - 72
18d = -27
d = -27/18
d = -1.5

Now that we know the common difference (d = -1.5), we can substitute this value back into equation (3) to find the first term (a):

a = 6 - 4(-1.5)
a = 6 + 6
a = 12

Therefore, the first term (a) is 12 and the common difference (d) is -1.5.

oops - that's a+4d=6

But I'm sure you caught that...

just crank it out using the formulas you already know:

a+5d = 6
12/2 (2a+11d) = 45