Question

How to find the third term in a sequence formed by adding the corresponding terms of a geometric sequence and an arithmetic sequence?

Answer 1

To find the third term in a sequence formed by adding corresponding terms of a geometric sequence and an arithmetic sequence, you need to know the specific details of both sequences.

Let's define the geometric and arithmetic sequences:

1. **Geometric Sequence**:
A geometric sequence can typically be expressed in terms of its first term \( a \) and common ratio \( r \). The \( n \)-th term of the geometric sequence is given by:
\[
G_n = a \cdot r^{n-1}
\]

2. **Arithmetic Sequence**:
An arithmetic sequence can be defined by its first term \( b \) and common difference \( d \). The \( n \)-th term of the arithmetic sequence is:
\[
A_n = b + (n-1) \cdot d
\]

3. **Combining the Sequences**:
The sequence formed by adding the corresponding terms of the geometric sequence and the arithmetic sequence will have each term defined as:
\[
S_n = G_n + A_n
\]
Thus,
\[
S_n = (a \cdot r^{n-1}) + (b + (n-1) \cdot d)
\]

### To find the third term \( S_3 \):
Substituting \( n = 3 \) into the formula:
\[
S_3 = G_3 + A_3
\]

Calculating the third terms:
\[
G_3 = a \cdot r^{3-1} = a \cdot r^2
\]
\[
A_3 = b + (3-1) \cdot d = b + 2d
\]

Thus, the third term \( S_3 \) is:
\[
S_3 = (a \cdot r^2) + (b + 2d)
\]

In conclusion, to find the third term in the sequence formed by adding the corresponding terms of a geometric sequence and an arithmetic sequence, you simply plug in the values for \( a \), \( r \), \( b \), and \( d \) into the equation:
\[
S_3 = a \cdot r^2 + b + 2d
\]