$$ k+1 ; k-1 ; 2 k-5 ; \ldots $$ are the first three terms of a geometric sequence, where $$ k

Question

$$ k+1 ; k-1 ; 2 k-5 ; \ldots $$ are the first three terms of a geometric sequence, where $$ k< $$ (a) Calculate the value of $$ k $$. (b) Determine the sequence. (c) Determine the general term of the sequence. (d) Find the 10 th term.

UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined:

### Step 1: Extract Known Conditions
We have the first three terms of a geometric sequence:
- First term: $$ a_1 = k + 1 $$
- Second term: $$ a_2 = k - 1 $$
- Third term: $$ a_3 = 2k - 5 $$

### Step 2: Solve for $$ k $$
In a geometric sequence, the ratio of consecutive terms is constant. Therefore, we can set up the following equation based on the ratios:

$$
\frac{a_2}{a_1} = \frac{a_3}{a_2}
$$

Substituting the terms:

$$
\frac{k - 1}{k + 1} = \frac{2k - 5}{k - 1}
$$

Now, we will cross-multiply to solve for $$ k $$:

$$
(k - 1)^2 = (k + 1)(2k - 5)
$$

Expanding both sides:

$$
k^2 - 2k + 1 = 2k^2 - 5k + 2k - 5
$$

This simplifies to:

$$
k^2 - 2k + 1 = 2k^2 - 3k - 5
$$

Rearranging gives:

$$
0 = 2k^2 - 3k - 5 - k^2 + 2k - 1
$$

Combining like terms:

$$
0 = k^2 - k - 6
$$

Now, we will factor the quadratic equation:

$$
0 = (k - 3)(k + 2)
$$

Setting each factor to zero gives us:

$$
k - 3 = 0 \quad \Rightarrow \quad k = 3
$$
$$
k + 2 = 0 \quad \Rightarrow \quad k = -2
$$

Since we have the condition $$ k < 0 $$, we take $$ k = -2 $$.

### Step 3: Determine the Sequence
Now substituting $$ k = -2 $$ into the terms:

- First term: 
$$
a_1 = k + 1 = -2 + 1 = -1
$$
- Second term: 
$$
a_2 = k - 1 = -2 - 1 = -3
$$
- Third term: 
$$
a_3 = 2k - 5 = 2(-2) - 5 = -4 - 5 = -9
$$

Thus, the sequence is:
$$
-1, -3, -9
$$

### Step 4: Determine the General Term of the Sequence
The common ratio $$ r $$ can be calculated as follows:

$$
r = \frac{a_2}{a_1} = \frac{-3}{-1} = 3
$$

The general term of a geometric sequence can be expressed as:

$$
a_n = a_1 \cdot r^{n-1}
$$

Substituting $$ a_1 = -1 $$ and $$ r = 3 $$:

$$
a_n = -1 \cdot 3^{n-1}
$$

### Step 5: Find the 10th Term
To find the 10th term, we substitute $$ n = 10 $$:

$$
a_{10} = -1 \cdot 3^{10-1} = -1 \cdot 3^9
$$

Calculating $$ 3^9 $$:

$$
3^9 = 19683
$$

Thus,

$$
a_{10} = -19683
$$

### Final Answers
(a) $$ k = -2 $$

(b) The sequence is $$ -1, -3, -9 $$

(c) The general term is $$ a_n = -1 \cdot 3^{n-1} $$

(d) The 10th term is $$ -19683 $$
(a) $$ k = -2 $$ (b) The sequence is $$ -1, -3, -9 $$ (c) The general term is $$ a_n = -3^{n-1} $$ (d) The 10th term is $$ -19683 $$

are the first three terms of a geometric sequence, where
(a) Calculate the value of .
(b) Determine the sequence.
© Determine the general term of the sequence.
(d) Find the 10 th term.

Upstudy AI Solution

Answer

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Beyond the Answer

(a) Calculate the value of .

(b) Determine the sequence.

© Determine the general term of the sequence.

(d) Find the 10th term.

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are the first three terms of a geometric sequence, where (a) Calculate the value of . (b) Determine the sequence. © Determine the general term of the sequence. (d) Find the 10 th term.