Answer:
a) 4n - 3
b) 6m + 5
Step-by-step explanation:
To answer these questions, let's break down the steps:
(a) Find an expression, in terms of [tex]n[/tex], for the [tex]n[/tex]th term of the given arithmetic sequence.
The given sequence is: 1, 5, 9, 13, 17.
Observing the sequence, we can identify that each term increases by a constant difference of 4. Therefore, this sequence is an arithmetic sequence with a common difference [tex]d = 4[/tex].
To find the [tex]n[/tex]th term ([tex]a_n[/tex]) of an arithmetic sequence, we use the formula:
[tex]\Large\boxed{\boxed{ a_n = a_1 + (n-1) \times d }}[/tex]
where:
- [tex] a_1 [/tex] is the first term of the sequence,
- [tex] d [/tex] is the common difference,
- [tex] n [/tex] is the term number.
In our sequence:
- [tex] a_1 = 1 [/tex] (the first term),
- [tex] d = 4 [/tex] (the common difference).
Plug these values into the formula:
[tex] a_n = 1 + (n-1) \times 4 [/tex]
[tex] a_n = 1 + 4(n-1) [/tex]
[tex] a_n = 1 + 4n - 4 [/tex]
[tex] a_n = 4n - 3 [/tex]
Therefore, the expression for the [tex]n[/tex]th term of this arithmetic sequence is [tex] \boxed{4n - 3} [/tex].
(b) Find an expression, in terms of [tex]m[/tex], for the [tex]2m[/tex]th term of another arithmetic sequence, where the [tex]m[/tex]th term is [tex]3n + 5[/tex].
Given the [tex]m[/tex]th term ([tex]a_m[/tex]) of the sequence is [tex]3n + 5[/tex], we need to find the expression for the [tex]2m[/tex]th term ([tex]a_{2m}[/tex]).
To find [tex]a_{2m}[/tex], we use the formula for the [tex]m[/tex]th term:
[tex] a_m = 3n + 5 [/tex]
Now, let's find [tex]a_{2m}[/tex] using [tex]n= 2m[/tex]:
[tex] a_{2m} = 3(2m)+ 5 [/tex]
[tex] a_{2m} = 6m+ 5 [/tex]
Therefore, the expression for the [tex]2m[/tex]th term of this arithmetic sequence is [tex] \boxed{6m + 5} [/tex].
