The nth term, un, of a geometric sequence is given by un = 3 (4)^n +1, n ∈ Z^+.
(a) Find the common ratio r.
(b) Hence, or otherwise, Find Sn, the sum of the first n terms of this sequence.
the ratio must be
(3*4^(n+1) + 1) / (3*4^n + 1)
But since that depends on n, you must have a typo
Sn = 4^(n+1) + n - 4
To find the common ratio, we can use the formula:
un = u1 * r^(n-1)
where u1 is the first term and r is the common ratio.
In this case, u1 = u0 = 3(4)^0 + 1 = 3 + 1 = 4
Now, let's substitute the values into the formula:
un = 4 * r^(n-1)
And we know that un = 3(4)^n + 1, so we can substitute that as well:
3(4)^n + 1 = 4 * r^(n-1)
Let's solve for r:
3(4)^n + 1 = 4 * r^(n-1)
Divide both sides by 4:
3(4)^(n-1) + 1/4 = r^(n-1)
Now, take the n-1th root of both sides to isolate r:
r = (3(4)^(n-1) + 1/4)^(1/(n-1))
This is the common ratio.
For part (b), to find the sum of the first n terms Sn of this geometric sequence, we can use the formula:
Sn = (u1 * (1 - r^n)) / (1 - r)
Let's substitute the values into the formula:
Sn = (4 * (1 - r^n)) / (1 - r)
We already found the value of r in part (a), so we can substitute it into the formula:
Sn = (4 * (1 - ((3(4)^(n-1) + 1/4)^(1/(n-1))))^n) / (1 - ((3(4)^(n-1) + 1/4)^(1/(n-1))))
This is the sum of the first n terms of the sequence.
To find the common ratio, r, of the geometric sequence, we can use the formula:
r = u_{n+1} / u_n
(a) Let's plug in the given values into the formula:
u_{n+1} = 3 * (4)^(n+1) + 1
u_n = 3 * (4)^n + 1
Now we can substitute these values into the formula to find the common ratio:
r = (3 * (4)^(n+1) + 1) / (3 * (4)^n + 1)
Simplifying the expression further:
r = (3 * 4^(n+1) + 1) / (3 * 4^n + 1)
Using exponent rules, we can rewrite 4^(n+1) as (4^n * 4^1):
r = (3 * 4^n * 4^1 + 1) / (3 * 4^n + 1)
Simplifying again:
r = (12 * 4^n + 1) / (3 * 4^n + 1)
Now we have found the common ratio, r.
(b) To find the sum of the first n terms, Sn, of this sequence, we can use the formula:
Sn = (u_1 * (1 - r^n)) / (1 - r)
Where u_1 is the first term of the sequence.
In this case, u_1 = u_1 = 3 * (4)^1 + 1 = 15.
Now, we can substitute the values into the formula to find Sn:
Sn = (15 * (1 - r^n)) / (1 - r)
We already found the common ratio, r, in part (a). Now we can substitute it into the formula to get the final expression for Sn.