NUMBER

Section B: seats $249$ to $249+315-1=563$. Last seat number $=563$.

Since $249 \le 400 \le 563$, seat 400 is in Section B, at position $400-249+1=152$ (the 152nd seat of Section B).

Pairing gives 25 pairs, each summing to $1+50=51$. Total $=25\times51=1275$.

There are 25 even numbers (2 to 50). Pairing first+last: $2+50=52$, giving $12.5$ pairs... more directly: sum $= 2(1+2+\cdots+25) = 2\times\frac{25\times26}{2}=650$.

$1+2+3=6$. $1+2+3+4=10$. $1+2+3+4+5=15$.

$n=3$: $\frac{3\times4}{2}=6$ ?. $n=4$: $\frac{4\times5}{2}=10$ ?. $n=5$: $\frac{5\times6}{2}=15$ ?. All match.

$$\frac{100\times101}{2}=5050$$ Since the formula matched exactly for every tested case, and it is a well-known proven result (Gauss's method), this prediction can be trusted with high confidence.

Writing $S = 1+2+\cdots+n$ and also $S=n+(n-1)+\cdots+1$, then adding term-by-term: each pair sums to $(n+1)$, and there are $n$ such pairs, giving $2S=n(n+1)$.

$$S = \frac{n(n+1)}{2}$$

$$\frac{12\times13}{2}=78 \text{ cans}$$

Testing: 13 rows needs $\frac{13\times14}{2}=91$ cans. 14 rows needs $\frac{14\times15}{2}=105$ cans (too many). So 13 complete rows can be built, using 91 cans, leaving $100-91=9$ cans left over.

$$u_{22} = 18+(22-1)(3) = 18+63=81 \text{ seats}$$

Total seats $=\frac{22}{2}(18+81)=11(99)=1089$. Reserved seats $=22\times2=44$. Occupied seats $=1089-44=1045$.

$$n+(n+1)+(n+2)+(n+3) = 4n+6$$

$4n+6 = 4(n+1)+2$ — this shows $4n+6$ is always 2 MORE than a multiple of 4 (i.e. it leaves remainder 2 when divided by 4), for every possible value of $n$. Since the remainder is never 0, the sum can never be exactly divisible by 4.

Sum $=7+8+9+10=34$. $34\div4=8.5$ (remainder 2, matching the proof) — confirms 34 is not divisible by 4.

Let $\mathbb{N} = \{1,2,3,4,\ldots\}$ denote the natural numbers (some definitions include 0, but the traditional MYP convention excludes it), and let $W=\{0,1,2,3,\ldots\}$ denote the whole numbers. The key distinction is that $0 \in W$ but $0 \notin \mathbb{N}$ under this convention — whole numbers include everything natural numbers do, PLUS zero.

Example: if you're COUNTING the number of students absent today, the answer must be a natural number if at least someone is being 'counted' as present in a meaningful group sense in some contexts — but more clearly, if you're recording a BANK BALANCE or TEMPERATURE, zero is a completely valid and meaningful value, which is why whole numbers (including 0) are needed there, while purely 'counting' contexts (like counting people in a room, which logically starts from 1 if the room isn't empty) sometimes naturally align with natural numbers.

$u_n = 240+(n-1)(35)$. Week 8: $u_8=240+7(35)=240+245=485$ riders.

$u_{20}=240+19(35)=240+665=905$ riders. This linear extrapolation is likely UNREALISTIC over such a long timeframe — real ridership growth typically slows and plateaus as it approaches the practical capacity of the route/buses, is affected by seasonal variation, and cannot increase forever at a constant rate. The model is likely only valid for the initial growth phase, not for long-term prediction.

$$u_{22} = 18+(22-1)(3) = 18+63=81 \text{ seats}$$

Total seats $=\frac{22}{2}(18+81)=11(99)=1089$. Reserved seats $=22\times2=44$. Occupied seats $=1089-44=1045$.

$$n+(n+1)+(n+2)+(n+3) = 4n+6$$

$4n+6 = 4(n+1)+2$ — this shows $4n+6$ is always 2 MORE than a multiple of 4 (i.e. it leaves remainder 2 when divided by 4), for every possible value of $n$. Since the remainder is never 0, the sum can never be exactly divisible by 4.

Sum $=7+8+9+10=34$. $34\div4=8.5$ (remainder 2, matching the proof) — confirms 34 is not divisible by 4.

Let $\mathbb{N} = \{1,2,3,4,\ldots\}$ denote the natural numbers (some definitions include 0, but the traditional MYP convention excludes it), and let $W=\{0,1,2,3,\ldots\}$ denote the whole numbers. The key distinction is that $0 \in W$ but $0 \notin \mathbb{N}$ under this convention — whole numbers include everything natural numbers do, PLUS zero.

Example: if you're COUNTING the number of students absent today, the answer must be a natural number if at least someone is being 'counted' as present in a meaningful group sense in some contexts — but more clearly, if you're recording a BANK BALANCE or TEMPERATURE, zero is a completely valid and meaningful value, which is why whole numbers (including 0) are needed there, while purely 'counting' contexts (like counting people in a room, which logically starts from 1 if the room isn't empty) sometimes naturally align with natural numbers.

$u_n = 240+(n-1)(35)$. Week 8: $u_8=240+7(35)=240+245=485$ riders.

$u_{20}=240+19(35)=240+665=905$ riders. This linear extrapolation is likely UNREALISTIC over such a long timeframe — real ridership growth typically slows and plateaus as it approaches the practical capacity of the route/buses, is affected by seasonal variation, and cannot increase forever at a constant rate. The model is likely only valid for the initial growth phase, not for long-term prediction.