Eric thinks of 2 sequences.One is geometric and the other arithmetic.Both sequences start with the number 3.The common ratio of the geometric sequence is the same as the common difference of the arithmetic sequence.If the 6-th term of the geometric sequence is 96.Find the first 5 terms of the arithmetic

we can get CD by getting the CR using geometric sequence since the common ratio (CR) & common diff (CD)are the same... so,

A sub 6=A sub 1 r^(n-1)
96=3r^5
r^5=32
r=2-------> CD is also 2

THEN,
the first 5 terms of an arithmetic sequence using CD of 2 are:
3, 5, 7, 9, 11

I know this is 6 years late but whatever

Using the guess and check method I did 3 times 3 but that was too much so I did 2 times 3 until I got to 96 which was the 6th term so if I do 3 plus 2 I can get 3, 5, 7, 9, 11, 14,

To find the first 5 terms of the arithmetic sequence, we need to determine the common difference.

We know that the 6th term of the geometric sequence is 96. Let's denote the common ratio of the geometric sequence as 'r'. Given that both sequences start with the number 3, we can write the 6th term of the geometric sequence as:

3 * r^5 = 96

To find the value of 'r', we solve this equation:

r^5 = 96/3

r^5 = 32

Taking the 5th root of both sides:

r = 2

Now that we have the value of 'r', which is the common difference of the arithmetic sequence, we can find the first 5 terms by adding 'r' to each preceding term.

Starting from 3, the first term of both sequences, the first 5 terms of the arithmetic sequence are:

3, 3+2, 3+2+2, 3+2+2+2, 3+2+2+2+2

Simplifying each term, we have:

3, 5, 7, 9, 11

Therefore, the first 5 terms of the arithmetic sequence are 3, 5, 7, 9, and 11.

To find the first 5 terms of the arithmetic sequence, we need to determine the common ratio of the geometric sequence.

We know that the common ratio of the geometric sequence is the same as the common difference of the arithmetic sequence. Let's assume this common difference/common ratio is represented by the variable 'd'.

Since both sequences start with the number 3, we can write the terms of the geometric sequence and the arithmetic sequence as follows:

Geometric sequence: 3, 3d, 3d^2, 3d^3, 3d^4, …

Arithmetic sequence: 3, 3+d, 3+2d, 3+3d, 3+4d, …

Given that the 6th term of the geometric sequence is 96, we can set up the equation:

3d^5 = 96

Now, we can solve for 'd'.

1. Divide both sides of the equation by 3 to isolate d^5:
d^5 = 96/3
d^5 = 32

2. Take the 5th root of both sides to isolate d:
d = ∛32

3. Simplify:
d = 2

Now that we know d = 2, we can substitute it back into the terms of the arithmetic sequence to find the first 5 terms:

Arithmetic sequence: 3, 3+2, 3+2(2), 3+3(2), 3+4(2)

Simplifying:
Arithmetic sequence: 3, 5, 7, 9, 11

Therefore, the first 5 terms of the arithmetic sequence are 3, 5, 7, 9, 11.