Please notice that an arithmetic sequence looks like this:
[tex]a_1,a_1+d,a_2+d,...,a_{n-1}+d \\ a_1+d,a_1+2d,a_1+3d,...,a_n(n-1)d[/tex]
Look at the terms:
1st term
2nd term = 1st term + d
3rd term = 2nd term + d = 1st term + 2d
4th term = 3rd term + d = 1st term + 3d
...
Observe that:
nth term = 1st term + (n-1)d
In order to prove this by induction,
When n=1
1st term ≟ 1st term + (1-1)d
1st term = 1st term + 0
1st term = 1st term TRUE
We let n=k
kth term = 1st term + (k-1) d , we know that this is true
But is this true for n=k+1?
(k+1)th term ≟ 1st term + (k+1-1)d
(k+1)th term ≟ 1st term + kd
We know for a fact that (k+1)th term = kth term + d
kth term + d ≟ 1st term + kd
Using the equation above,
1st term + (k-1)d + d ≟ 1st term + kd
(k-1+1)d ≟ kd
kd = kd TRUE
