A sequence is formed by adding together the corresponding terms of a geometric sequence and and an arithmetic sequence.The common ratio of the geometric sequence is 2 and the common difference of the arithmetic sequence is 2.The first term of the new sequence is 1 and the second term is 7.

Could you please help me find the third term of the new sequence.
Thank you.

AS: a+2n

GS: b*2^(n-1)

a+b = 1
a+2 + 2b = 7

a + b = 1
a + 2b = 5
b = 4
a = -3

-3 + 2n: -3 -1 1 3 5
4*2^(n-1): 4 8 16 32 64
sum: 1 7 17 35 69

Looks like 17

Sure! To find the third term of the new sequence, we need to understand the pattern of how the terms are formed.

Let's break down the problem step by step.

First, let's find the first term of both the geometric sequence and the arithmetic sequence.

Since the first term of the new sequence is given as 1, it must be the sum of the first terms of the geometric and arithmetic sequences. Let's call the first term of the geometric sequence "a" and the first term of the arithmetic sequence "d".

So, we have the equation:
1 = a + d

Next, let's find the second term of both sequences.

The second term of the new sequence is given as 7, which is also the sum of the second terms of the geometric and arithmetic sequences.

In a geometric sequence, the second term is obtained by multiplying the first term by the common ratio. Therefore, the second term of the geometric sequence is 2a.

In an arithmetic sequence, the second term is obtained by adding the common difference to the first term. Therefore, the second term of the arithmetic sequence is a + d.

So, we have the equation:
7 = 2a + (a + d)

Now, we have a system of two equations:
1 = a + d
7 = 2a + (a + d)

Let's solve this system of equations to find the values of "a" and "d".

Subtract the first equation from the second equation:
7 - 1 = 2a - a + d - d
6 = a

Now that we have the value of "a", let's substitute it back into the first equation to find the value of "d":
1 = 6 + d
d = -5

So, the first term of the geometric sequence is 6 (a = 6) and the first term of the arithmetic sequence is -5 (d = -5).

Now that we know the values of "a" and "d", we can find the third term of the new sequence.

The third term is obtained by adding the third term of the geometric sequence and the third term of the arithmetic sequence.

In a geometric sequence, the third term is obtained by multiplying the first term by the common ratio squared. Therefore, the third term of the geometric sequence is 2a^2.

In an arithmetic sequence, the third term is obtained by adding twice the common difference to the first term. Therefore, the third term of the arithmetic sequence is a + 2d.

Let's calculate the third term of the new sequence:
Third term = 2a^2 + (a + 2d)
= 2(6)^2 + (6 + 2(-5))
= 2(36) + (6 - 10)
= 72 - 4
= 68

Therefore, the third term of the new sequence is 68.

I hope this explanation helps you understand how to solve the problem! Let me know if you have any further questions.