We will investigate whether it is possible to have more than one arithmetic series with four terms whose sum is $44 .$ Let's first recall the definition of an arithmetic series.

The sum of the terms in an arithmetic sequence is called an arithmetic series.

According to this definition, it looks like we should focus on the summation of the terms. Let's now recall the formula for the sum of an arithmetic series.

S_{n} = n (\frac{a _{1} + a _{n}}{2})

By substituting the given values

n

=

4

and

S_{4}

=

44

in the above formula, we can find a condition that the desired arithmetic series must satisfy.

S_{n} = n (\frac{a _{1} + a _{n}}{2})

SubstituteII

$n = 4$ , $S_{n} = 44$

44 = 4 (\frac{a _{1} + a _{4}}{2})

DivEqn

$LHS / 4 = RHS / 4$

11 = \frac{a _{1} + a _{4}}{2}

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

22 = a_{1} + a_{4}

RearrangeEqn

Rearrange equation

a_{1} + a_{4} = 22

With this information, we can conclude that any arithmetic series with

4

terms in which the sum of the first and fourth terms is

22,

will have a sum of

44 .

Let's write three examples. Recall that the difference between consecutive terms must be constant!

Example series 1 : Example series 2 : Example series 3 : 2 + 8 + 14 + 20 5 + 9 + 13 + 17 - 1 + 7 + 15 + 23

The above are examples of arithmetic series because they have a constant common difference. Moreover, note that the sum of the four terms of each series is

44 .

Also, note that the sum of the first and fourth terms is

22 .

Exercise

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