Arithmetic and Geometric Series

An old legend states that a peasant won a reward from a king. The peasant asked to be paid in rice; one grain on the first square of a chessboard, two grains on the second, four on the third square, and so on. Which square would contain exactly $512$ grains of rice?

An old legend states that a peasant won a reward from a king. The peasant asked to be paid in rice; one grain on the first square of a chessboard, two grains on the second, four on the third square, and so on. How many grains of rice would be on the ${30}^{\mathrm{th}}$ square?

The first three terms of an infinite geometric sequence are $m-1, 6, m+8$. Calculate the sum to infinity.

The first three terms of an infinite geometric sequence are $m-1, 6, m+8$. Find two possible values of $r$. Only one of these $r$ values forms a geometric sequence where an infinite sum can be found. Justify your choice for $r$.

The first three terms of an infinite geometric sequence are $m-1, 6, m+8$. Find two possible values of $r$.

The first three terms of an infinite geometric sequence are $m-1, 6, m+8$. Find two possible values of $m$.

The first three terms of an infinite geometric sequence are $m-1, 6, m+8$. Write down two expressions for $r$.

A geometric sequence has all positive terms. The sum of the first two terms is $15$ and the sum to infinity is $27$. Find the first term.

A geometric sequence has all positive terms. The sum of the first two terms is $15$ and the sum to infinity is $27$. Find the value of the common ratio.

A large company created a phone tree to contact all employees in case of an emergency. Each of the five vice presidents calls five employees, who in turn each call five other employees, and so on. How many rounds of phone calls are needed to reach all $2375$ employees?

In a geometric sequence, the fourth term is $8$ times the first term. The sum of the first $10$ terms is $2557.5$. Find the $10$th term of this sequence.

Create two different infinite geometric series that each have a sum of 8.

The sum of an infinite geometric series is three times the first term. Find the common ratio of this series.

The sum of an infinite geometric series is $20$, and the common ratio is $0.2$. Find the first term of this series.

In an arithmetic series, $S_{40}=1900\text{ and }{u}_{40}=106$. Find the value of the first term and the common difference.

In an arithmetic sequence, the first term is $-8$ and the sum of the first $20$ terms is $790$. Find how many terms it takes for the sum to exceed $2000$

In an arithmetic sequence, the first term is $-8$ and the sum of the first $20$ terms is $790$. Find $S_{28}$

In an arithmetic sequence, the first term is $-8$ and the sum of the first $20$ terms is $790$. Find $u_8$

In an arithmetic sequence, the first term is $-8$ and the sum of the first $20$ terms is $790$. Find the common difference.

Find the sum of all Integers between $500$ and $1400$ (inclusive) that are not divisible by $7$.