Sub-sequences of an arithmetic sequence

A sub-sequence is a sequence made of some of the terms of a sequence, in their order. In the sequence $\displaystyle 5, 8, 11, 14, 17, 20, \ldots$ ($\displaystyle {d}={3}$):

• The terms in the odd places, $\displaystyle a_1, a_3, a_5, \ldots$: $\displaystyle 5, 11, 17, \ldots$ – between any two there are two steps of $\displaystyle d$, so this is an arithmetic sequence with difference $\displaystyle {2d}={6}$.
• The terms in the even places, $\displaystyle a_2, a_4, a_6, \ldots$: $\displaystyle 8, 14, 20, \ldots$ – arithmetic too, with difference $\displaystyle {2d}={6}$.
• Every third term, $\displaystyle a_1, a_4, a_7, \ldots$: difference $\displaystyle 3d$.

• In a sequence of 40 terms: 20 terms in the even places ($\displaystyle a_2, a_4, \ldots, a_{40}$), and 20 in the odd places.
• In a sequence of 41 terms: 21 in the odd places ($\displaystyle a_1, a_3, \ldots, a_{41}$), and 20 in the even places.

A hall has 16 rows.
The first row has 10 seats, and each row has 2 seats more than the row before it.
How many seats are there in the even rows?

• The even rows are 8 rows: the first of them has $\displaystyle {a_2}={12}$ seats, and the difference is $\displaystyle {2\cdot2}={4}$.
• $\displaystyle {S}={\frac{8(2\cdot12+7\cdot4)}{2}}={\frac{8\cdot52}{2}}={208}$ seats.

• Change the sign of every term: $\displaystyle 86, 82, \ldots, -70$ becomes $\displaystyle -86, -82, \ldots, 70$ – an arithmetic sequence with difference $\displaystyle -d$, whose sum is minus the sum of the first one.
• Add the same number to every term: the difference does not change.
• Multiply every term by a number $\displaystyle k$: the difference is multiplied by $\displaystyle k$.

An arithmetic sequence has $\displaystyle {a_1}={4}$, $\displaystyle {d}={5}$ and 80 terms.
The sum of the last 20 terms is the sum of them all minus the sum of the first 60, $\displaystyle S_{80}-S_{60}$:

• $\displaystyle {S_{80}}={\frac{80(8+79\cdot5)}{2}}={16120}$, and $\displaystyle {S_{60}}={\frac{60(8+59\cdot5)}{2}}={9090}$.
• $\displaystyle {S_{80}-S_{60}}={7030}$.

The sub-sequence $\displaystyle a_{61}, \ldots, a_{80}$ gives the same sum: $\displaystyle {\frac{20(304+399)}{2}}={7030}$.

The arithmetic sequence $\displaystyle 3, 7, 11, \ldots$ has 40 terms.

How many terms are in the even places, and what is the difference between them?

Calculate the sum of the terms in the even places.

An arithmetic sequence has $\displaystyle {a_1}={2}$, $\displaystyle {d}={3}$ and 29 terms.
Calculate the sum of the terms in the odd places.

The arithmetic sequence $\displaystyle 50, 47, 44, \ldots$ has 30 terms.
Calculate the sum of its last 10 terms.

The arithmetic sequence: $\displaystyle 86, 82, 78, \ldots, -70$ (40 terms).
The sign of every term is changed, giving a new sequence.
Calculate the sum of the new sequence.