To determine whether you have an arithmetic sequence, find the difference between the first few and the last few numbers. Ensure that the difference is always the same. For example, the series 10, 15, 20, 25, 30 is an arithmetic sequence, because the difference between each term is constant (5).
For example, if you are calculating the sum of the sequence 10, 15, 20, 25, 30, n=5{\displaystyle n=5}, since there are 5 terms in the sequence.
For example, in the sequence 10, 15, 20, 25, 30 a1=10{\displaystyle a_{1}=10}, and an=30{\displaystyle a_{n}=30}.
Note that this formula is indicating that the sum of the arithmetic sequence is equal to the average of the first and last term, multiplied by the number of terms. [3] X Research source
For example, if you have 5 terms in your sequence, and 10 is the first term, and 30 is the last term, your formula will look like this: Sn=5(10+302){\displaystyle S_{n}=5({\frac {10+30}{2}})}.
For example:Sn=5(402){\displaystyle S_{n}=5({\frac {40}{2}})}Sn=5(20){\displaystyle S_{n}=5(20)}
For example: Sn=5(20){\displaystyle S_{n}=5(20)}Sn=100{\displaystyle S_{n}=100}So, the sum of the sequence 10, 15, 20, 25, 30 is 100.
Determine the number of terms (n{\displaystyle n}) in the sequence. Since you are considering all consecutive integers to 500, n=500{\displaystyle n=500}. Determine the first (a1{\displaystyle a_{1}}) and last (an{\displaystyle a_{n}}) terms in the sequence. Since the sequence is 1 to 500, a1=1{\displaystyle a_{1}=1} and an=500{\displaystyle a_{n}=500}. Find the average of a1{\displaystyle a_{1}} and an{\displaystyle a_{n}}: 1+5002=250. 5{\displaystyle {\frac {1+500}{2}}=250. 5}. Multiply the average by n{\displaystyle n}: 250. 5×500=125,250{\displaystyle 250. 5\times 500=125,250}.
Determine the number of terms (n{\displaystyle n}) in the sequence. Since you begin with 3, end with 24, and go up by 7 each time, the series is 3, 10, 17, 24. (The common difference is the difference between each term in the sequence. )[7] X Research source This means that n=4{\displaystyle n=4} Determine the first (a1{\displaystyle a_{1}}) and last (an{\displaystyle a_{n}}) terms in the sequence. Since the sequence is 3 to 24, a1=3{\displaystyle a_{1}=3} and an=24{\displaystyle a_{n}=24}. Find the average of a1{\displaystyle a_{1}} and an{\displaystyle a_{n}}: 3+242=13. 5{\displaystyle {\frac {3+24}{2}}=13. 5}. Multiply the average by n{\displaystyle n}: 13. 5×4=54{\displaystyle 13. 5\times 4=54}.
Determine the number of terms (n{\displaystyle n}) in the sequence. Since Mara save for 52 weeks (1 year), n=52{\displaystyle n=52}. Determine the first (a1{\displaystyle a_{1}}) and last (an{\displaystyle a_{n}}) terms in the sequence. The first amount she saves is 5 dollars, so a1=5{\displaystyle a_{1}=5}. To find out the amount she saves the last week of the year, calculate 5×52=260{\displaystyle 5\times 52=260}. So an=260{\displaystyle a_{n}=260}. Find the average of a1{\displaystyle a_{1}} and an{\displaystyle a_{n}}: 5+2602=132. 5{\displaystyle {\frac {5+260}{2}}=132. 5}. Multiply the average by n{\displaystyle n}: 132. 5×52=6,890{\displaystyle 132. 5\times 52=6,890}. So she saves $6,890 by the end of the year.
