# Question 99b2b

Nov 8, 2016

The question only asks for the missing number which is 34

#### Explanation:

When faced with a sequence you first need to establish if is arithmetic or geometric.

By example

Arithmetic:
Progresses by some variant on addition or subtraction of a constant

Geometric:
Progresses by some variant on the multiplication or division of some constant raised to a power.

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Observe that:

$40 - 37 = 3$
$31 - 28 = 3$
$28 - 25 = 3$

This indicates that it is a arithmetic progression and that as it is reducing the change is $- 3$

color(green)("So the missing number is "37-3=34#

Check: $34 - 3 = 31 \textcolor{red}{\leftarrow \text{Confirmed}}$
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$\textcolor{b l u e}{\text{As this is the algebra section - Developing the equation}}$

Let the first term be ${a}_{1} = 40$

Let the ${i}^{\text{th}}$ term be ${a}_{i}$

${a}_{1} = {a}_{1} - 0 = 40$
${a}_{2} = {a}_{1} - 3 = 37$
${a}_{3} = {a}_{1} - 3 - 3 = 34$
${a}_{4} = {a}_{1} - 3 - 3 - 3 = 31$

By observation we deduce that:

$\textcolor{b l u e}{\implies {a}_{i} = {a}_{1} - 3 \left(i - 1\right) = 40 - 3 \left(i - 1\right)}$

check:
$\implies {a}_{1} = 40 - 3 \left(1 - 1\right) = 40 - 0 = 40$