Arithmetic Sequences & Series

[Maximum mark: 6]

Emily starts reading Leo Tolstoy's War and Peace on the $1$st of February. The number of pages she reads each day increases by the same number on each successive day.

1. Calculate the number of pages Emily reads on the $14$th of February. [3]
   

2. Find the exact total number of pages Emily reads in the $28$ days of February.[3]
   

[Maximum mark: 6]

A battalion is arranged, per row, according to an arithmetic sequence. There are $24$ troops in the third row and $42$ troops in the sixth row.

1. Find the first term and the common difference of this arithmetic sequence. [3]
   

There are $15$ rows in the battalion.

2. Find the total number of troops in the battalion. [3]
   

[Maximum mark: 12]

The sum of the first $n$ terms of an arithmetic sequence, $S_n = u_1 + u_2 + u_3 + \dots + u_n$, is given by $S_n = 2n^2 + n$.

1. Write down the values of $S_1$ and $S_2$. [2]
   

2. Write down the values of $u_1$ and $u_2$. [2]
   

3. Find $d$, the common difference of the sequence. [1]
   

4. Find $u_{10}$, the tenth term of the sequence. [2]
   

5. Find the greatest value of $n$, for which $u_n$ is less than $100$. [3]
   

6. Find the value of $n$, for which $S_n$ is equal to $1275$. [2]
   

[Maximum mark: 6]

Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is $16.1$ km. The first milestone placed on the road is $124.7$ km from London, and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is $667.1$ km.

1. Find the distance between the fifth milestone and London. [3]
   

2. Determine how many milestones there are along the road. [3]