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Sequences and Series

What's the sum of the first 100 positive integers? How about the first 1000?

Sequences Warmup


If \(\{a_n\}\) is a sequence defined by \(a_n = n^2 + n + 1\) for each natural number \(n\), what is \(a_5?\)

Suppose \(\{a_n\}\) is a sequence defined by \[a_1 = 1, a_2 = 1,\] and \[a_n = a_{n-1} + a_{n -2}\] for each natural number \(n > 2.\)

What is \(a_6\)?

A geometric progression is a sequence in which \(a_n = r \cdot a_{n-1}\) for each natural number \(n > 1\), where \(r\) is a real number called the common ratio.

If \(a_n\) is a geometric progression with \(a_1 = 5\) and \(a_6 = 160\), what is \(a_3\)?

On Day 1, Isabel has $200 in the bank, and she adds $5 at the start of each subsequent day. On what day will her account’s value reach $300?

Each day this week, Morgan had twice as much money as the day before. On Day 6, Morgan had $40. How many dollars did Morgan have on Day 1?


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