Question

the seven term of arithemetic sequence 25inthe first third and 21sth term of geometric sequence find the first term and the common diffrence sequence

Answer 1

To solve this problem, we need to clarify the conditions provided for the arithmetic and geometric sequences. Let's denote:

- \( a_n \) as the nth term of the arithmetic sequence.
- \( g_n \) as the nth term of the geometric sequence.

For an arithmetic sequence, the nth term can be expressed as:

\[
a_n = a + (n-1)d
\]

where \( a \) is the first term and \( d \) is the common difference.

For a geometric sequence, the nth term can be expressed as:

\[
g_n = b \cdot r^{(n-1)}
\]

where \( b \) is the first term and \( r \) is the common ratio.

### Given Information:
1. The 7th term of the arithmetic sequence is 25:
\[
a_7 = a + 6d = 25 \quad (1)
\]

2. The 3rd and 21st terms of the geometric sequence:
\[
g_3 = b \cdot r^2 \quad (2)
\]
\[
g_{21} = b \cdot r^{20} \quad (3)
\]

We will assume both the 3rd and 21st terms of the geometric sequence equal the value of 25. This creates two conditions:

\[
g_3 = g_{21} \quad (4)
\]

Substituting from (2) and (3):
\[
b r^2 = b r^{20}
\]

Assuming \( b \neq 0 \) (since a term cannot be zero), we can simplify:
\[
r^2 = r^{20}
\]

Dividing both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[
1 = r^{18}
\]

This gives us two cases:
1. \( r = 1 \)
2. \( r = -1 \)

### Case 1: \( r = 1 \)

If \( r = 1 \):
- The geometric sequence becomes a constant sequence where every term equals \( b \).
- From (2): \( b = 25 \)

Thus, all terms are \( 25 \).

### Case 2: \( r = -1 \)

If \( r = -1 \):
- The geometric sequence alternates between \( b \) and \( -b \).
- For odd \( n \), \( g_n = b \); for even \( n \), \( g_n = -b \).

Thus:
- \( g_3 = 25 \Rightarrow b = 25 \)
- \( g_{21} = 25 \Rightarrow b = 25 \)

So we might still proceed with \( b = 25 \). The terms alternate due to \( r = -1 \).

### Finding the Arithmetic Sequence
Now we can return to our arithmetic sequence:

From equation (1):
\[
a + 6d = 25 \quad (5)
\]

Since we have no additional constraints, we choose values for \( d \) and solve for \( a \).

#### Example Selection:
1. If we let \( d = 0 \):
\[
a + 6(0) = 25 \Rightarrow a = 25
\]
Therefore, the arithmetic sequence could simply be a constant sequence of 25.

2. If we let \( d = 1 \):
\[
a + 6(1) = 25 \Rightarrow a + 6 = 25 \Rightarrow a = 19
\]
The sequence would be: \( 19, 20, 21, 22, 23, 24, 25, \ldots \)

In conclusion, there are multiple solutions for the arithmetic sequence based on different choices of the common difference \( d \). The significant parameters you need to double-check based on the contexts of the sequences are:

- First term of arithmetic sequence \( a \)
- Common difference \( d \)

If you have any specific values for the geometric sequence's terms, please specify to refine the solutions.