Are the vectors u = i + 2j and v = 2i + 4j collinear?

v = 2u, so the two vectors are collinear (parallel).

No, the dot product is non-zero

Yes, but only because they are perpendicular

No, they have different magnitudes

What is the sum of the infinite geometric series 1 + (1)/(2) + (1)/(4) + …?

Geometric series with a=1, r=(1)/(2). Sum = (a)/(1-r) = (1)/(1-1/2) = 2.

The sum telescopes: Σ_k=1^99 ((1)/(k) - (1)/(k+1)) = 1 - (1)/(100) = (99)/(100).

Problems & worked solutions

RO M1

Are the vectors $\vec{u} = \vec{i} + 2 \vec{j}$ and $\vec{v} = 2 \vec{i} + 4 \vec{j}$ collinear?

Show answer & worked solution

$\vec{v} = 2 \vec{u}$ , so the two vectors are collinear (parallel).

RO M1

What is the sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \dots$ ?

Show answer & worked solution

Geometric series with $a = 1$ , $r = \frac{1}{2}$ . Sum $= \frac{a}{1 - r} = \frac{1}{1 - 1 / 2} = 2$ .

International

$\sum_{k = 1}^{99} \frac{1}{k (k + 1)}$ equals:

Show answer & worked solution

The sum telescopes: $\sum_{k = 1}^{99} (\frac{1}{k} - \frac{1}{k + 1}) = 1 - \frac{1}{100} = \frac{99}{100}$ .

Problems & worked solutions

RO M1

Are the vectors $\vec{u} = \vec{i} + 2 \vec{j}$ and $\vec{v} = 2 \vec{i} + 4 \vec{j}$ collinear?

Show answer & worked solution

$\vec{v} = 2 \vec{u}$ , so the two vectors are collinear (parallel).

RO M1

What is the sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \dots$ ?

Show answer & worked solution

Geometric series with $a = 1$ , $r = \frac{1}{2}$ . Sum $= \frac{a}{1 - r} = \frac{1}{1 - 1 / 2} = 2$ .

International

$\sum_{k = 1}^{99} \frac{1}{k (k + 1)}$ equals:

Show answer & worked solution

The sum telescopes: $\sum_{k = 1}^{99} (\frac{1}{k} - \frac{1}{k + 1}) = 1 - \frac{1}{100} = \frac{99}{100}$ .