Sure! Let's go through the given tasks step by step.
### Part (a): Writing the arithmetic sequence.
We are given:
- The first term (\(a_1\)) is 1.
- The fifth term (\(a_5\)) is 1.04.
In an arithmetic sequence, each term is derived by adding a constant value, called the common difference (\(d\)), to the previous term. The general formula for the \(n\)th term (\(a_n\)) of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
For the fifth term:
[tex]\[ a_5 = a_1 + 4d \][/tex]
Given:
[tex]\[ a_1 = 1 \][/tex]
[tex]\[ a_5 = 1.04 \][/tex]
Let's substitute the given values into the formula for the fifth term:
[tex]\[ 1.04 = 1 + 4d \][/tex]
Now, solve for the common difference (\(d\)):
[tex]\[ 1.04 - 1 = 4d \][/tex]
[tex]\[ 0.04 = 4d \][/tex]
[tex]\[ d = \frac{0.04}{4} = 0.01 \][/tex]
Thus, the common difference (\(d\)) is 0.01.
Now we can write the first five terms of the sequence:
- First term: \(a_1 = 1\)
- Second term: \(a_2 = a_1 + d = 1 + 0.01 = 1.01\)
- Third term: \(a_3 = a_1 + 2d = 1 + 2 \times 0.01 = 1.02\)
- Fourth term: \(a_4 = a_1 + 3d = 1 + 3 \times 0.01 = 1.03\)
- Fifth term: \(a_5 = a_1 + 4d = 1 + 4 \times 0.01 = 1.04\)
So, the arithmetic sequence with the first term as 1 and the fifth term as 1.04 is:
[tex]\[ 1, 1.01, 1.02, 1.03, 1.04 \][/tex]
### Part (b): Finding the 10th term.
To find the 10th term (\(a_{10}\)), we use the formula for the nth term of an arithmetic sequence:
[tex]\[ a_{10} = a_1 + (10 - 1)d \][/tex]
Substituting the values:
[tex]\[ a_{10} = 1 + 9 \times 0.01 \][/tex]
[tex]\[ a_{10} = 1 + 0.09 \][/tex]
[tex]\[ a_{10} = 1.09 \][/tex]
Therefore, the 10th term ([tex]\(a_{10}\)[/tex]) of the sequence is 1.09.
