The sum 1 + 2 + 3 + … + 100 equals:

Apply the formula (n(n+1))/(2) = (100 · 101)/(2) = 5050.

The limit lim_x → 1 (x^3 - 3x + 2)/(x - 1) equals (apply Bezout / factoring):

x^3 - 3x + 2 = (x - 1)(x^2 + x - 2) = (x - 1)(x - 1)(x + 2) = (x - 1)^2 (x + 2). So ((x-1)^2(x+2))/(x - 1) = (x - 1)(x + 2) → 0 as x → 1.

Daily Math Problems — June 5, 2026: Sequences and series, Combinatorics, Calculus

dailymath

Problems & worked solutions

International

Problem 1 — Sequences

The sum $1 + 2 + 3 + \dots + 100$ equals:

Show answer & worked solution

A. $5050$ ✓ correct
B. $5000$
C. $10000$
D. $4950$

Apply the formula $\frac{n (n + 1)}{2} = \frac{100 \cdot 101}{2} = 5050$ .

RO M1

Problem 2 — Binomial Theorem

The middle term of the expansion of $(x + 1)^{8}$ has coefficient:

Show answer & worked solution

A. $28$
B. $56$
C. $70$ ✓ correct
D. $128$

Middle coefficient: $(\binom{8}{4}) = 70$ .

RO M1

Problem 3 — Limits of Functions

The limit $\lim_{x \to 1} \frac{x^{3} - 3 x + 2}{x - 1}$ equals (apply Bezout / factoring):

Show answer & worked solution

A. $0$ ✓ correct
B. $1$
C. $3$
D. $\infty$

$x^{3} - 3 x + 2 = (x - 1) (x^{2} + x - 2) = (x - 1) (x - 1) (x + 2) = (x - 1)^{2} (x + 2)$ . So $\frac{(x - 1)^{2} (x + 2)}{x - 1} = (x - 1) (x + 2) \to 0$ as $x \to 1$ .

Problems & worked solutions

International

Problem 1 — Sequences

The sum $1 + 2 + 3 + \dots + 100$ equals:

Show answer & worked solution

A. $5050$ ✓ correct
B. $5000$
C. $10000$
D. $4950$

Apply the formula $\frac{n (n + 1)}{2} = \frac{100 \cdot 101}{2} = 5050$ .

RO M1

Problem 2 — Binomial Theorem

The middle term of the expansion of $(x + 1)^{8}$ has coefficient:

Show answer & worked solution

A. $28$
B. $56$
C. $70$ ✓ correct
D. $128$

Middle coefficient: $(\binom{8}{4}) = 70$ .

RO M1

Problem 3 — Limits of Functions

The limit $\lim_{x \to 1} \frac{x^{3} - 3 x + 2}{x - 1}$ equals (apply Bezout / factoring):

Show answer & worked solution

A. $0$ ✓ correct
B. $1$
C. $3$
D. $\infty$

$x^{3} - 3 x + 2 = (x - 1) (x^{2} + x - 2) = (x - 1) (x - 1) (x + 2) = (x - 1)^{2} (x + 2)$ . So $\frac{(x - 1)^{2} (x + 2)}{x - 1} = (x - 1) (x + 2) \to 0$ as $x \to 1$ .