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Consider all consecutive integers.  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}. The first term in the sequence is 3. The last term in the sequence is 24. The common difference is 7.  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.) 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}. Mara saves 5 dollars the first week of the year. For the rest of the year, she increases her weekly savings by 5 dollars every week. How much money does Mara save by the end of the year?  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.

Summary:
Find the sum of numbers between 1 and 500. Find the sum of the described arithmetic sequence. Solve the following problem.