Mohamad and Vanessa have both recently started their own small businesses, and neither of them initially have any clients. Over the first year, Mohamad's client base increased by $$17 per month, while Vanessa's client base increased by $$5 in its first month, $$10 in its second month, $$15 in its third month, and so on.

Sure, let's break down the problem step by step.

1. Mohamad's Client Base Growth:

   • Mohamad's client base increases by $$17 per month.
   • This means that after 1 month, he has $$17 clients.
   • After 2 months, he has $$17 \times 2 = $$34 clients.
   • After 3 months, he has $$17 \times 3 = $$51 clients.
   • This pattern continues linearly.

2. Vanessa's Client Base Growth:

   • Vanessa's client base increases in an arithmetic sequence.
   • In the first month, she gains $$5 clients.
   • In the second month, she gains $$10 clients.
   • In the third month, she gains $$15 clients.
   • This means that in the nth month, she gains $$5n clients.

3. Calculating Total Clients for Vanessa:

   • To find the total number of clients Vanessa has after a certain number of months, we need to sum the arithmetic sequence.
   • The sum of the first n terms of an arithmetic sequence where the first term is a and the common difference is d is given by: $S_n = \frac{n}{2} \times (2a + (n-1)d)$
   • For Vanessa, $a = 5$ and $d = 5$.
   • Therefore, the sum of clients after n months is: $S_n = \frac{n}{2} \times (2 \times 5 + (n-1) \times 5) = \frac{n}{2} \times (10 + 5n - 5) = \frac{n}{2} \times (5n + 5) = \frac{n}{2} \times 5(n + 1) = \frac{5n(n + 1)}{2}$

4. Comparing Client Bases:

   • To find when Vanessa's client base equals or exceeds Mohamad's, we set their client base equations equal to each other: $17n = \frac{5n(n + 1)}{2}$
   • Solving for n: $34n = 5n(n + 1)$ $34n = 5n^2 + 5n$ $5n^2 - 29n = 0$ $n(5n - 29) = 0$
   • This gives us $n = 0$ or $n = \frac{29}{5} = 5.8$.

5. Conclusion:

   • Since n must be a whole number, we round up to the next whole number, which is 6.
   • Therefore, after 6 months, Vanessa's client base will exceed Mohamad's client base.