How and what do I assume for my induction? Hi I wrote a formula to know the 100th term for the following sequence:
$$3, 10, 17, 24, 31$$
Note that each following number in the sequence will be 7 more than the previous number.
And here is the formula:
$$7N - 4$$
$$
\begin{array}{c|c}
Term   &amp; 1 &amp; 2  &amp; 3  &amp; 4  &amp; 5  &amp; Nth       &amp; 100 \\ \hline
Number &amp; 3 &amp; 10 &amp; 17 &amp; 24 &amp; 31 &amp; 7\times N - 4 &amp; 696 \\
\end{array}
$$
I proved it with the base case $$5 \times 7 - 4 = 31$$
which is correct.
And I proved it with$$100\times7 - 4 = 696,$$
which is correct.
Also I was looking at this induction example:


So that I can apply on my own induction. But the problem is I can't really apply it, because I don't know what to assume/claim to prove it. This is where I'm stuck:
$$
\mathrm{Induction \enspace step;  \enspace Assume  \enspace true  \enspace for  \enspace n=k,  \enspace show \enspace true \enspace n=k+1}
$$
$$
\mathrm{Assume: 3,10,17,24,...,[WHAT \enspace DO \enspace I \enspace NEED \enspace TO \enspace WRITE \enspace HERE?] = 7N-4}
$$
Does someone has any idea what I need to write in the above assumption? Thanks in advance.
The induction step will be trivial.
Assuming the $k$th term is given by $7k+4$, the $(k+1)$th term is $(7k+4) + 7 = 7(k+1)+4$.
This completes the induction.
Let $a_n$ be the $n$'th term in an arithmetic sequence, where:
$$
a_1 = 3
$$
$$
d = 7
$$
We find the $n+1$ term as:
$$
a_{n+1} = 3 + 7n
$$
If the the sequence is arithmetic, then :
$$
a_{n} = a_{n+1} - 7    
$$
$$
\downarrow
$$
$$
a_{n}+7 = a_{n+1}
$$
Now we can see from the third row and the last row:
$$
a_{n+1}= a_n + 7
$$
$$
a_{n+1} = 3 + 7n
$$
$$
\downarrow
$$
$$
a_n + 7 = 3 + 7n
$$
$$
a_n = 7n-4
$$