Operations on an arithmetic sequence and sub-sequences

$\displaystyle a_n$ is an arithmetic sequence with difference $\displaystyle d$. These sequences are arithmetic too:

• $\displaystyle a_n+3$: each term grows by 3, and the difference stays $\displaystyle d$. The graph moves 3 units up.
• $\displaystyle 3a_n$: the difference is $\displaystyle {3a_{n+1}-3a_n}={3d}$. The graph is stretched by 3 from the $\displaystyle x$-axis.
• $\displaystyle a_{n+2}$: the same sequence, from its third term. The difference is $\displaystyle d$, and the graph moves 2 units left.
• $\displaystyle a_{2n}$: the terms in the even places, $\displaystyle a_2, a_4, a_6, \ldots$ Between them are two steps, so the difference is $\displaystyle 2d$.

The proof in every case: the difference between two neighbouring terms of the new sequence is the same number for every $\displaystyle n$.

The sequence $\displaystyle a_n^2$ is usually not arithmetic. A counterexample: $\displaystyle 1, 2, 3$ gives $\displaystyle 1, 4, 9$, and the differences $\displaystyle 3$ and $\displaystyle 5$ are not equal.
One counterexample is enough to show that the claim “$\displaystyle a_n^2$ is always arithmetic” is false.

A sub-sequence is some of the terms of a sequence, in their order. In the arithmetic sequence $\displaystyle 3, 7, 11, 15, 19, 23, \ldots$ ($\displaystyle {d}={4}$):

• The terms in the odd places: $\displaystyle 3, 11, 19, \ldots$ – an arithmetic sequence with difference $\displaystyle 8$.
• The terms in the even places: $\displaystyle 7, 15, 23, \ldots$ – an arithmetic sequence with difference $\displaystyle 8$.
• The sum of the terms in the odd places among the first 20 terms: 10 terms, $\displaystyle {\frac{10\left[2\cdot3+9\cdot8\right]}{2}}={390}$.

The sequence $\displaystyle 1, 10, 3, 8, 5, 6, 7, 4, \ldots$ is not arithmetic, but it interleaves two arithmetic sequences:

• in the odd places $\displaystyle 1, 3, 5, 7, \ldots$ – difference $\displaystyle 2$;
• in the even places $\displaystyle 10, 8, 6, 4, \ldots$ – difference $\displaystyle -2$.

The sum of the first 20 terms is the sum of 10 terms of each sequence:

• $\displaystyle {\frac{10\left[2\cdot1+9\cdot2\right]}{2}}={100}$;
• $\displaystyle {\frac{10\left[2\cdot10+9\cdot(-2)\right]}{2}}={10}$;
• together: $\displaystyle 110$.

$\displaystyle a_n$ is an arithmetic sequence with difference $\displaystyle {d}={4}$.
Which of these sequences are arithmetic? For each arithmetic one write its difference, and for each other one – a counterexample.

• $\displaystyle a_n-7$
• $\displaystyle -2a_n$
• $\displaystyle a_{3n}$
• $\displaystyle \frac{a_n}{5}$
• $\displaystyle a_n^2$

$\displaystyle {a_n}={a_1+(n-1)d}$ is an arithmetic sequence, and $\displaystyle {b_n}={a_{2n}}$.
Simplify $\displaystyle b_{n+1}-b_n$, and explain what it proves.

The sequence: $\displaystyle 50, 1, 47, 4, 44, 7, \ldots$ (one arithmetic sequence in the odd places, and another in the even places).

Calculate $\displaystyle a_{30}$.

Calculate $\displaystyle a_{31}$.

Calculate the sum of the first 30 terms.

An arithmetic sequence has 40 terms.
The sum of the terms in the odd places is 600, and the sum of the terms in the even places is 640.
Find $\displaystyle a_1$ and $\displaystyle d$.