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.