The 10th term of an arithmetic sequence is 42 more than the 7th term.If the 1st term is -14,find the second term of the sequence
The two terms are three apart, so
3d = 42
d = 14
So now
a+d = -14+14 = 0
To find the second term of the arithmetic sequence, we need to first determine the common difference.
Given the information that the 10th term is 42 more than the 7th term, we can set up the following equation:
10th term = 7th term + 42
Using the formula for the nth term of an arithmetic sequence, we have:
a + (n-1)d = a + (m-1)d + 42
Since the 1st term, a, is -14, we substitute this value into the equation:
-14 + 9d = -14 + 6d + 42
Next, we simplify the equation:
9d - 6d = 42
3d = 42
To solve for d, we divide both sides of the equation by 3:
d = 14
Now that we have found the common difference, we can find the second term, which is one position ahead of the first term. To do this, we add the common difference to the first term:
-14 + 14 = 0
Therefore, the second term of the sequence is 0.
To find the second term of the arithmetic sequence, we need to determine the common difference first. Here's how you can do it:
1. Identify the given information:
- 10th term: This is the term with index 10 in the sequence, denoted as a10.
- 7th term: This is the term with index 7 in the sequence, denoted as a7.
- 1st term: This is the term with index 1 in the sequence, denoted as a1.
2. Use the formula for finding the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
Substituting the given values for the 10th term and 7th term, we get:
a10 = a1 + 9d
a7 = a1 + 6d
3. Set up the equation:
The 10th term is 42 more than the 7th term, so we can write the equation as:
a10 = a7 + 42
4. Substitute the expressions from step 2 into the equation:
a1 + 9d = a1 + 6d + 42
5. Simplify and solve for d:
Subtract a1 from both sides:
9d = 6d + 42
Subtract 6d from both sides:
3d = 42
Divide both sides by 3:
d = 14
6. Find the second term using the formula:
a2 = a1 + d
Substitute the given value for the 1st term:
a2 = -14 + 14
Simplify:
a2 = 0
Therefore, the second term of the arithmetic sequence is 0.