Prove using strong induction that there exists an integer $C$ that for all $D \ge C$ it is possible to return exactly $D$ by using summed composites of only $2$ and $5$
so far i believe that $\exists C$, $\forall D$ $P(c,d)$ is true $\implies d = 2c+5c$, but i am not sure if thats the correct approach. Once I understand the base step then the process becomes more simplified. But as of now, I am unable to get beyond this point. I am unable to understand what it wants,
my base step would be
$P(1,7)---> 7 = 2(1)+5(1)$ is $c\le d$, yes $1 \le 7$ but i am having doubts on the next step following this
does the question want a logically reasoned methodology? what does it want?
As I understood from your comment, you're trying to prove that any positive integer $d$ can be written as $d=2a+5b$ where $a$ and $b$ are also positive integers.
If this is the question then the answer is yes for $d > 3$.
If $d$ is even then $d=2k$ so take $a=k$ and $b=0$
If $d$ is odd then $d=2k+1$ with $k\ge 2$, so for example if $d=5$ then it is true $d=2(0)+5(1)$, now let's suppose it is true up to $d'$ (i.e. $\exists a',b' \ge 0$ such that $d'=2a'+5b'$) and lets prove that the statement is true for $d= d'+2$ (since $d$ is also odd), $d=d'+2=2a'+5b'+2= 2(a'+1)+5b'$, so $a=a'+1$ and $b=b'$