A new sequence is formed by adding together the corresponding terms of a geometric sequence and an arithmetric sequence. the geometric sequence has a common ratio of 3 and the arthmetric sequence has a common difference of -2.

The first two terms of the new sequence are 4 and 20.

how do you calculate the third term of the new sequence?

What is the 21st term of the sequence with a1 = -5 and d = 6?

To calculate the third term of the new sequence, we need to add the corresponding terms of the geometric and arithmetic sequences.

Let's first find the first term of each sequence:

The first term of the geometric sequence is given as 4.

The first term of the arithmetic sequence can be found by subtracting the common difference from the second term of the new sequence:
20 - (-2) = 20 + 2 = 22.

Now, let's find the second term of each sequence:

The second term of the geometric sequence is obtained by multiplying the common ratio (3) with the first term:
4 * 3 = 12.

The second term of the arithmetic sequence can be found by adding the common difference to the first term of the arithmetic sequence:
22 + (-2) = 22 - 2 = 20.

To calculate the third term of the new sequence, we add the corresponding terms of the geometric and arithmetic sequences:

Third term = (Third term of geometric sequence) + (Third term of arithmetic sequence)

The formula for the nth term of a geometric sequence is given by: a * r^(n - 1), where a is the first term and r is the common ratio.

Similarly, the formula for the nth term of an arithmetic sequence is given by: a + (n - 1) * d, where a is the first term and d is the common difference.

Let's calculate the values:

Third term of geometric sequence = 4 * (3)^(3 - 1) = 4 * 9 = 36.

Third term of arithmetic sequence = 22 + (3 - 1) * (-2) = 22 - 4 = 18.

Now, we can find the third term of the new sequence by adding the terms:

Third term of new sequence = Third term of geometric sequence + Third term of arithmetic sequence
= 36 + 18
= 54.

Therefore, the third term of the new sequence is 54.

To calculate the third term of the new sequence, you need to find the sum of the third terms of the geometric and arithmetic sequences. Here's how you can do it:

1. First, let's write down the formulas for the geometric and arithmetic sequences:
- For the geometric sequence: Aᵢ = A₁ * r^(i-1)
- For the arithmetic sequence: Aᵢ = A₁ + (i-1) * d

Where Aᵢ represents the i-th term of the sequence, A₁ is the first term, r is the common ratio, and d is the common difference.

2. Since the given geometric sequence has a common ratio of 3, we can write down its formula using A₁ = 4:
- Geometric sequence: Aᵢ = 4 * 3^(i-1)

3. Also, the given arithmetic sequence has a common difference of -2, so its formula becomes:
- Arithmetic sequence: Aᵢ = 20 + (i-1) * (-2)

4. Now, to find the third term of the new sequence, we add the third terms of the geometric and arithmetic sequences:
- Third term of the new sequence: A₃ = (4 * 3^(3-1)) + (20 + (3-1) * (-2))

5. Calculate the exponentiation and multiplication first:
- Third term of the new sequence: A₃ = (4 * 9) + (20 + 2 * (-2))

6. Simplify the expression:
- Third term of the new sequence: A₃ = 36 + (20 - 4)

7. Continue simplifying:
- Third term of the new sequence: A₃ = 36 + 16

8. Calculate the final result:
- Third term of the new sequence: A₃ = 52

So, the third term of the new sequence is 52.

I will assume that you are adding the corresponding terms of each sequence, that is ..

new sequence:
t1 = a+a
t2 = a+d + ar
t3 = a+2d + ar^2
etc

so a+a = 4
a+d + ar = 20
but d=-2 and r=3

2a = 4
a = 2
So all is known

t3 = 2 - 4 + 2(9) = 16

check:
AS: 2, 0, -2, -4, ..
GS: 2, 6, 18, 54, ..

new sequence is :
4, 6, 16 , 50 ..
All looks good