Example 2
The common difference of an arithmetic sequence is 10 and the 5 th term is 49 . Find the general formula for the sequence.

Example 3
In an arithmetic sequence, the 2 nd term is 1 and the 7 th term is 16 . Find the 1st three terms of the sequence.
2) In an arithmetic sequence, and . Determine the first term and the constant difference.
3) Find the number of terms in the arithmetic sequence

The first three terms of an arithmetic sequence are, . Determine
a) The value of .
b) The general term.
c) The value of the term

Example 4
Consider the following arithmetic sequence:

(a) Find the value of .
(b) Find the 1st three terms of the sequence.
© Find .

Ask by Watkins Beck. in South Africa

Jan 21,2025

Question

Example 2
The common difference of an arithmetic sequence is 10 and the 5 th term is 49 . Find the general formula for the sequence.

Example 3
In an arithmetic sequence, the 2 nd term is 1 and the 7 th term is 16 . Find the 1st three terms of the sequence.
2) In an arithmetic sequence, and . Determine the first term and the constant difference.
3) Find the number of terms in the arithmetic sequence

The first three terms of an arithmetic sequence are, . Determine
a) The value of .
b) The general term.
c) The value of the term

Example 4
Consider the following arithmetic sequence:

(a) Find the value of .
(b) Find the 1st three terms of the sequence.
© Find .

Ask by Watkins Beck. in South Africa

Jan 21,2025

UpStudy AI · Accepted Answer

Let's solve each example step by step.

### Example 2
**Problem:** The common difference of an arithmetic sequence is 10 and the 5th term is 49. Find the general formula for the sequence.

**Known Conditions:**
- Common difference $$ d = 10 $$
- 5th term $$ T_5 = 49 $$

**Step 1:** The general formula for the $$ n $$-th term of an arithmetic sequence is given by:
$$
T_n = a + (n-1)d
$$
where $$ a $$ is the first term.

**Step 2:** For the 5th term:
$$
T_5 = a + 4d
$$
Substituting the known values:
$$
49 = a + 4(10)
$$
$$
49 = a + 40
$$
$$
a = 49 - 40 = 9
$$

**Step 3:** Now we can write the general formula:
$$
T_n = 9 + (n-1) \cdot 10
$$
$$
T_n = 9 + 10n - 10
$$
$$
T_n = 10n - 1
$$

### Example 3
**Problem 1:** In an arithmetic sequence, the 2nd term is 1 and the 7th term is 16. Find the first three terms of the sequence.

**Known Conditions:**
- $$ T_2 = 1 $$
- $$ T_7 = 16 $$

**Step 1:** Using the general formula:
$$
T_n = a + (n-1)d
$$
For the 2nd term:
$$
T_2 = a + d = 1 \quad (1)
$$
For the 7th term:
$$
T_7 = a + 6d = 16 \quad (2)
$$

**Step 2:** Subtract equation (1) from equation (2):
$$
(a + 6d) - (a + d) = 16 - 1
$$
$$
5d = 15 \implies d = 3
$$

**Step 3:** Substitute $$ d $$ back into equation (1):
$$
a + 3 = 1 \implies a = -2
$$

**Step 4:** The first three terms are:
$$
T_1 = a = -2, \quad T_2 = 1, \quad T_3 = a + 2d = -2 + 6 = 4
$$
Thus, the first three terms are $$ -2, 1, 4 $$.

**Problem 2:** In an arithmetic sequence, $$ T_3 = -2 $$ and $$ T_8 = 23 $$. Determine the first term and the constant difference.

**Known Conditions:**
- $$ T_3 = a + 2d = -2 \quad (1) $$
- $$ T_8 = a + 7d = 23 \quad (2) $$

**Step 1:** Subtract equation (1) from equation (2):
$$
(a + 7d) - (a + 2d) = 23 + 2
$$
$$
5d = 25 \implies d = 5
$$

**Step 2:** Substitute $$ d $$ back into equation (1):
$$
a + 2(5) = -2 \implies a + 10 = -2 \implies a = -12
$$

### Example 4
**Problem:** Consider the following arithmetic sequence: $$ 2k-1, k+7, 5k-5 $$.

**Step 1:** Set the common differences equal:
$$
(k + 7) - (2k - 1) = (5k - 5) - (k + 7)
$$
Simplifying both sides:
$$
k + 7 - 2k + 1 = 5k - 5 - k - 7
$$
$$
- k + 8 = 4k - 12
$$
$$
8 + 12 = 4k + k \implies 20 = 5k \implies k = 4
$$

**Step 2:** Find the first three terms:
$$
T_1 = 2(4) - 1 = 8 - 1 = 7
$$
$$
T_2 = 4 + 7 = 11
$$
$$
T_3 = 5(4) - 5 = 20 - 5 = 15
$$
Thus, the first three terms are $$ 7, 11, 15 $$.

**Step 3:** The general term $$ T_n $$:
$$
T_n = a + (n-1)d
$$
Where $$ a = 7 $$ and $$ d = 4 $$:
$$
T_n = 7 + (n-1) \cdot 4
$$
$$
T_n = 7 + 4n - 4 = 4n + 3
$$

### Summary of Results
- Example 2: $$ T_n = 10n - 1 $$
- Example 3 (1): First three terms are $$ -2, 1, 4 $$
- Example 3 (2): First term $$ a = -12 $$, common difference $$ d = 5 $$
- Example 4: $$ k = 4 $$, first three terms $$ 7, 11, 15 $$, general term $$ T_n = 4n + 3 $$
- **Example 2:** The general formula for the sequence is $$ T_n = 10n - 1 $$. - **Example 3 (1):** The first three terms are $$ -2, 1, 4 $$. - **Example 3 (2):** The first term is $$ -12 $$ and the common difference is $$ 5 $$. - **Example 4 (a):** The value of $$ k $$ is $$ 4 $$. - **Example 4 (b):** The first three terms are $$ 7, 11, 15 $$. - **Example 4 (c):** The general term is $$ T_n = 4n + 3 $$.

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