How to Find the Sum of Additives

An arithmetic progression is a sequence of numbers in which each term increases by a constant amount. To sum the terms of an arithmetic progression, you can add all the numbers mentally. However, this will not be possible when the arithmetic progression includes multiple terms. Instead, you can quickly find the sum of the arithmetic progression by multiplying the average of the first and last terms by the number of terms.

Table of Contents

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Make sure you have an arithmetic progression. An arithmetic progression is a sequence of consecutive numbers in which the difference between the terms is constant. ^{[1] X Research Source} This method is only correct if your series of numbers is additive.

To determine if you have an arithmetic progression, find the difference between the top few terms in the series and between the terms at the bottom. Make sure that difference doesn’t change.
For example, the sequence 10, 15, 20, 25, 30 is an additive because the difference between consecutive terms is constant (5).

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Calculate the number of terms in the arithmetic progression. Each number in the arithmetic progression is called a term. If there are only a few terms then you can count. On the other hand, if you know the first term, the last term, and the variance (the difference between each term), you can use the formula to calculate the number of terms. Let’s set the number of terms to be searched as the variable

n

{displaystyle n}

n

.

Suppose if you calculate the sum of the arithmetic progression 10, 15, 20, 25, 30 then $n$ $=$ $5$ ${displaystyle n=5}$ $n=5$ , because there are 5 terms in the arithmetic progression.

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Determine the first and last terms of the arithmetic progression. You need to know both of these terms to calculate the sum of the arithmetic progressions. Usually the first term will be 1 but not always. Let’s set the variable

a

first

{displaystyle a_{1}}

a_{{1}}

is the first term of the arithmetic progression and

a

n

{displaystyle a_{n}}

a_{{n}}

is the last term of the arithmetic progression.

For example, in the arithmetic progression 10, 15, 20, 25, 30, $a$ $first$ $=$ $ten$ ${displaystyle a_{1}=10}$ $a_{{1}}=10$ , and $a$ $n$ $=$ $30$ ${displaystyle a_{n}=30}$ $a_{{n}}=30$ .

Total

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Write the formula for the sum of the arithmetic progressions. The formula would be:

S

n

=

n

(

a

first

+

a

n

2

)

{displaystyle S_{n}=n({frac {a_{1}+a_{n}}{2}})}

S_{{n}}=n({frac {a_{{1}}+a_{{n}}}{2}})

, in there

S

n

{displaystyle S_{n}}

SN}}

is the sum of the arithmetic progression. ^{[2] X Research Source}

Note that this formula shows that the sum of the arithmetic progressions is equal to the average of the first and last terms multiplied by the terms. ^{[3] X Research Sources}

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Substitute the values of $n$ ${displaystyle n}$ $n$ , $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ , and $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ into the formula. Make sure you change the correct number.

For example, if you have 5 terms in the arithmetic progression, 10 being the first, and 30 being the last, the formula would look like this: $S$ $n$ $=$ $5$ $($ $ten$ $+$ $30$ $2$ $)$ ${displaystyle S_{n}=5({frac {10+30}{2}})}$ $S_{{n}}=5({frac {10+30}{2}})$ .

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Calculate the average of the first and last terms. To do this calculation, add the two numbers above and divide by 2.

For example:
$S$ $n$ $=$ $5$ $($ $40$ $2$ $)$ ${displaystyle S_{n}=5({frac {40}{2}})}$ $S_{{n}}=5({frac {40}{2}})$
$S$ $n$ $=$ $5$ $($ $20$ $)$ ${displaystyle S_{n}=5(20)}$ $S_{{n}}=5(20)$

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Multiply the average of two numbers by the number of terms in the series. You will get the sum of the arithmetic progression.

For example:
$S$ $n$ $=$ $5$ $($ $20$ $)$ ${displaystyle S_{n}=5(20)}$ $S_{{n}}=5(20)$
$S$ $n$ $=$ $100$ ${displaystyle S_{n}=100}$ $S_{{n}}=100$
Thus, the sum of the arithmetic progression 10, 15, 20, 25, 30 is 100.

Complete sample problems

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Find the sum of the numbers 1 to 500. Treat these numbers as consecutive integers.

Determine the number of terms ( $n$ ${displaystyle n}$ $n$ ) in the arithmetic progression. Since we are treating as a sequence of consecutive integers up to 500, $n$ $=$ $500$ ${displaystyle n=500}$ $n=500$ .
Determine the first term ( $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ ) and the last term ( $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ ) in the arithmetic progression. Since the arithmetic progression is a sequence of numbers from 1 to 500, so $a$ $first$ $=$ $first$ ${displaystyle a_{1}=1}$ $a_{{1}}=1$ and $a$ $n$ $=$ $500$ ${displaystyle a_{n}=500}$ $a_{{n}}=500$ .
Find the average of $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ and $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ : $first$ $+$ $500$ $2$ $=$ $250$ $,$ $5$ ${displaystyle {frac {1+500}{2}}=250.5}$ ${frac {1+500}{2}}=250.5$ .
Multiply the average plus $n$ ${displaystyle n}$ $n$ : $250$ $,$ $5$ $\times$ $500$ $=$ $125,250$ ${displaystyle 250.5times 500=125,250}$ $250.5times 500=125.250$ .

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Calculate the sum of the arithmetic progression, where the first term is 3, the last term is 24, and the difference is 7.

Find the number of terms ( $n$ ${displaystyle n}$ $n$ ) in the arithmetic progression. Since the first term you have is 3, the last term is 24, and each number is 7 units apart, the sequence will be 3, 10, 17, 24. (Correct difference is the difference between each term in the arithmetic progression. ). ^{[4] X Research Source} That means $n$ $=$ $4$ ${displaystyle n=4}$ $n=4$
Find the first term ( $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ ) and the last term ( $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ ) of the arithmetic progression. Since the arithmetic progression is a sequence of numbers from 3 to 24, so $a$ $first$ $=$ $3$ ${displaystyle a_{1}=3}$ $a_{{1}}=3$ and $a$ $n$ $=$ $24$ ${displaystyle a_{n}=24}$ $a_{{n}}=24$ .
Calculate the average of $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ and $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ : $3$ $+$ $24$ $2$ $=$ $13$ $,$ $5$ ${displaystyle {frac {3+24}{2}}=13.5}$ ${frac {3+24}{2}}=13.5$ .
Multiply the average plus $n$ ${displaystyle n}$ $n$ : $13$ $,$ $5$ $\times$ $4$ $=$ $54$ ${displaystyle 13.5times 4=54}$ $13.5times 4=54$ .

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Solve the following problem. Mara saves $5 in the first week of the year. For the rest of the year, she increases her weekly savings by $5 per week. How much money can Mara save at the end of the year?

Find the number of terms ( $n$ ${displaystyle n}$ $n$ ) in the arithmetic progression. Since Mara saves for 52 weeks (1 year), so $n$ $=$ $52$ ${displaystyle n=52}$ $n=52$ .
Find the first term ( $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ ) and the last term ( $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ ) of the arithmetic progression. Initial savings is 5 dollars, so $a$ $first$ $=$ $5$ ${displaystyle a_{1}=5}$ $a_{{1}}=5$ . To find the amount saved in the last week of the year, we do the math $5$ $\times$ $52$ $=$ $260$ ${displaystyle 5times 52=260}$ $5times 52=260$ . So $a$ $n$ $=$ $260$ ${displaystyle a_{n}=260}$ $a_{{n}}=260$ .
Calculate the average of $a$ $first$ ${displaystyle a_{1}}$ $a_{{1}}$ and $a$ $n$ ${displaystyle a_{n}}$ $a_{{n}}$ : $5$ $+$ $260$ $2$ $=$ $132$ $,$ $5$ ${displaystyle {frac {5+260}{2}}=132,5}$ ${frac {5+260}{2}}=132.5$ .
Multiply the average plus $n$ ${displaystyle n}$ $n$ : $135$ $,$ $5$ $\times$ $52$ $=$ $7.046$ ${displaystyle 135.5times 52=7,046}$ $135.5times 52=7,046$ . Thus, by the end of the year, Mara saved $7,046.

Steps

Rate your arithmetic progression

Total

Complete sample problems

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