Sure, let’s solve the given problem step by step.
The [tex]\( n^{\text{th}} \)[/tex] term, [tex]\( T_n \)[/tex], of the sequence is given by
[tex]\[ T_n = 3n - 1 \][/tex]
(a) The [tex]\( 5^{\text{th}} \)[/tex] term:
To find the [tex]\( 5^{\text{th}} \)[/tex] term, we substitute [tex]\( n = 5 \)[/tex] into the formula:
[tex]\[ T_5 = 3(5) - 1 \][/tex]
Carrying out the multiplication and subtraction:
[tex]\[ T_5 = 15 - 1 \][/tex]
[tex]\[ T_5 = 14 \][/tex]
Thus, the 5th term is [tex]\( 14 \)[/tex].
(b) The [tex]\( 12^{\text{th}} \)[/tex] term:
To find the [tex]\( 12^{\text{th}} \)[/tex] term, we substitute [tex]\( n = 12 \)[/tex] into the formula:
[tex]\[ T_{12} = 3(12) - 1 \][/tex]
Carrying out the multiplication and subtraction:
[tex]\[ T_{12} = 36 - 1 \][/tex]
[tex]\[ T_{12} = 35 \][/tex]
Thus, the 12th term is [tex]\( 35 \)[/tex].
(c) The difference between the [tex]\( 12^{\text{th}} \)[/tex] term and the [tex]\( 5^{\text{th}} \)[/tex] term:
To find the difference, we subtract the 5th term from the 12th term:
[tex]\[ \text{Difference} = T_{12} - T_5 \][/tex]
Substitute the values we obtained:
[tex]\[ \text{Difference} = 35 - 14 \][/tex]
Carrying out the subtraction:
[tex]\[ \text{Difference} = 21 \][/tex]
Thus, the difference between the 12th term and the 5th term is [tex]\( 21 \)[/tex].
