To find the required terms of the sequence defined by [tex]\( a_n = 2n - 1 \)[/tex], we will substitute the appropriate values of [tex]\( n \)[/tex] into the formula.
### Step 1: Find the Initial Term ([tex]\( a_1 \)[/tex])
First, we'll find [tex]\( a_1 \)[/tex] by substituting [tex]\( n = 1 \)[/tex] into the formula.
[tex]\[ a_1 = 2(1) - 1 \][/tex]
Simplifying this, we get:
[tex]\[ a_1 = 2 - 1 = 1 \][/tex]
So, [tex]\( a_1 = 1 \)[/tex].
### Step 2: Find the First 4 Terms
Next, we find the first four terms of the sequence. The first term, [tex]\( a_1 \)[/tex], is already known to be 1. Now we calculate the next three terms:
For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2(2) - 1 \][/tex]
[tex]\[ a_2 = 4 - 1 = 3 \][/tex]
For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 2(3) - 1 \][/tex]
[tex]\[ a_3 = 6 - 1 = 5 \][/tex]
For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 2(4) - 1 \][/tex]
[tex]\[ a_4 = 8 - 1 = 7 \][/tex]
So, the first four terms are [tex]\( 1, 3, 5, \)[/tex] and [tex]\( 7 \)[/tex].
### Step 3: Find the 10th Term ([tex]\( a_{10} \)[/tex])
We now find the 10th term by substituting [tex]\( n = 10 \)[/tex]:
[tex]\[ a_{10} = 2(10) - 1 \][/tex]
[tex]\[ a_{10} = 20 - 1 = 19 \][/tex]
So, [tex]\( a_{10} = 19 \)[/tex].
### Step 4: Find the 15th Term ([tex]\( a_{15} \)[/tex])
Finally, we find the 15th term by substituting [tex]\( n = 15 \)[/tex]:
[tex]\[ a_{15} = 2(15) - 1 \][/tex]
[tex]\[ a_{15} = 30 - 1 = 29 \][/tex]
So, [tex]\( a_{15} = 29 \)[/tex].
### Summary
- The first term, [tex]\( a_1 \)[/tex], is [tex]\( 1 \)[/tex].
- The first four terms of the sequence are [tex]\( 1, 3, 5, \)[/tex] and [tex]\( 7 \)[/tex].
- The 10th term, [tex]\( a_{10} \)[/tex], is [tex]\( 19 \)[/tex].
- The 15th term, [tex]\( a_{15} \)[/tex], is [tex]\( 29 \)[/tex].
Thus, the requested terms of the sequence are:
[tex]\[ a_1 = 1 \][/tex]
[tex]\[ \text{First 4 terms: } 1, 3, 5, 7 \][/tex]
[tex]\[ a_{10} = 19 \][/tex]
[tex]\[ a_{15} = 29 \][/tex]
