Answer:
Option 2nd is correct
[tex]a_n =8n-14[/tex]
Step-by-step explanation:
The nth term for the arithmetic equation is give by:
[tex]a_n = a_1+(n-1)d[/tex] ....[A]
where,
[tex]a_1[/tex] is the first term
d is the common difference and
n is the number of term.
As per the statement:
The fourth term in an arithmetic sequence is 18 and the seventh term is 42
⇒[tex]a_4 = 18[/tex] and [tex]a_7 = 42[/tex]
then by above definition we have;
[tex]a_1+3d = 18[/tex] .....[1]
[tex]a_1+6d = 42[/tex] .....[2]
Subtract equation [1] from [2] we have;
3d = 24
Divide both sides by 3 we have;
d = 8
Substitute in [1] we have;
[tex]a_1+24 = 18[/tex]
Subtract 24 from both sides we have;
[tex]a_1=-6[/tex]
We have to find the equation for the nth term of this sequence
Substitute the given values in [A] we have;
[tex]a_n = -6+(n-1)\cdot 8[/tex]
⇒[tex]a_n = -6+8n-8[/tex]
⇒[tex]a_n =8n-14[/tex]
Therefore, an equation for the nth term of this sequence is, [tex]a_n =8n-14[/tex]
