Arithmetic Progression

Katharina and carolina go to a swimming pool. they both swim the first length of the pool in $2$ minutes $6$ seconds. The time that katharina takes to swim each subsequent length is $5$ seconds more than she look to swim the previous length. The time that carolina takes to swim each subsequent length is $1.04$ times that which she took to swim the previous length.

katharina and carolina swim a total of $10$ length of the pool.they start at exactly at the same time.

Show that katharina completes the $10$ lengths before carolina.

Hence state the value of the time difference, correct to the nearest second, between their total times.

Katharina and carolina go to a swimming pool. they both swim the first length of the pool in $2$ minutes $6$ seconds. The time that katharina takes to swim each subsequent length is $5$ seconds more than she look to swim the previous length. The time that carolina takes to swim each subsequent length is $1.04$ times that which she took to swim the previous length.

katharina and carolina swim a total of $10$ length of the pool.they start at exactly at the same time.

Show that katharina completes the $10$ lengths before carolina.

Katharina and carolina go to a swimming pool. they both swim the first length of the pool in $2$ minutes $6$ seconds.

The time that katharina takes to swim each subsequent length is $5$ seconds more than she look to swim the previous length.

The time that carolina takes to swim each subsequent length is $1.04$ times that which she took to swim the previous length.

State the time katharina takes to swim the third length.

Mr williams is about to retire, and he needs to decide how his pension should be paid. His pension provider offers him three payment options:

option 1:$\mathrm{US} \$10000$ each year for $10$ years.

option 2:$\mathrm{US} \$2500$ in the first year,and then the amount increases by  $\mathrm{US} \$2000$ in each subsequent year for for a for a total of $10$ years.

option 3:$\mathrm{US} \$100$ in the first year, and then the amount doubles in each subsequent year for a for a total of $10$ years.

Determine which option has the greatest total value. Justify your answer by showing all appropriate calculations.

Consider the sum $S=\sum _{i=3}^x(3i+1)$, where $x$ is a positive integer greater than $3$.

Given that $S=520$, find the value of $x$.

$k^2,6+k,12-k$ are consecutive values of a sequence $\left\{a_n,n\ge 0\right\}$. Find the value of $k$ for which $\left\{a_n\right\}$ is a non-constant arithmetic sequence, stating its general term.

The $n$th term, $u_n$ of an arithmetic sequence is given by the formula $u_n=23+7n$.

 Explain why $1007$ is not a term in this sequence. 

The $n$th term, $u_n$ of an arithmetic sequence is given by the formula $u_n=23+7n$.

 State the $50$th term of the sequence.

The $n$th term, $u_n$ of an arithmetic sequence is given by the formula $u_n=23+7n$.

 State the first term of the sequence.

A sequence is given by $a_n=-10n+3$.

Show that this sequence is arithmetic.

Consider this sequence of figures.

Figure $1$ contains three line segments.

Find the total number of line segments in the first $100$ figures.

Consider this sequence of figures.

Figure $1$ contains three line segments.

Given that Figure $n$ contains $251$ line segments, show that $n=125$.

Consider this sequence of figures.

Figure $1$ contains three line segments.

Find the number of line segments in Figure $7$.

If $S_n$ is the sum of the first $n$ terms of the arithmetic series $105+100+95+\dots$

Find the value of $n$ for which $S_n=1140$.

Write down an expression for $S_n$ the sum of the first $n$ terms of the arithmetic series $105+100+95+\dots$

Consider the arithmetic series $S=20+20.4+20.8+\dots +37.6$

Find the value of $S$.

Consider the arithmetic series $S=20+20.4+20.8+\dots +37.6$

Find the number of terms in this series.

The first term of an arithmetic sequence is $u_1=5$ and the common difference is $d=2.5$.

 Find the the smallest value of $n$ for which $u_n>377$.

The first term of an arithmetic sequence is $u_1=5$ and the common difference is $d=2.5$.

Write down the general term $u_n$ of this sequence.

The first term of an arithmetic sequence is $u_1=5$ and the common difference is $d=2.5$.

Find $u_2$ and $u_3$.