A term's place and the number of terms

In the sequence $\displaystyle 7, 11, 15, \ldots$ ($\displaystyle {a_1}={7}$, $\displaystyle {d}={4}$) we find in which place the term 207 is: we put $\displaystyle {a_n}={207}$ into the general term, and solve an equation in $\displaystyle n$.

• Subtract 7: $\displaystyle {(n-1)\cdot4}={200}$.
• Divide by 4: $\displaystyle {n-1}={50}$, so $\displaystyle {n}={51}$.
• 207 is the 51st term of the sequence.

The place of a term is always a natural number. Is 100 a term of the same sequence?

• $\displaystyle {7+(n-1)\cdot4}={100}$ gives $\displaystyle {n-1}={23.25}$, that is $\displaystyle {n}={24.25}$.
• $\displaystyle 24.25$ is not a natural number, so 100 is not a term: it lies between $\displaystyle {a_{24}}={99}$ and $\displaystyle {a_{25}}={103}$.

The arithmetic sequence $\displaystyle 86, 82, 78, \ldots, -70$ ends at $\displaystyle -70$. The number of its terms is the place of its last term:

• $\displaystyle {a_1}={86}$, $\displaystyle {d}={82-86}={-4}$.
• $\displaystyle {86+(n-1)\cdot(-4)}={-70}$, so $\displaystyle {(n-1)\cdot(-4)}={-156}$, and $\displaystyle {n-1}={39}$.
• $\displaystyle {n}={40}$: the sequence has 40 terms.

In the same sequence, the positive terms are those with $\displaystyle a_n>0$:

• Expand the brackets: $\displaystyle 90-4n>0$, so $\displaystyle n<22.5$.
• $\displaystyle n$ is natural, so $\displaystyle {n}={1, 2, \ldots, 22}$: the sequence has 22 positive terms.
• The smallest positive term is the last of them: $\displaystyle {a_{22}}={86-21\cdot4}={2}$.

• The rule $\displaystyle {a_1}={4}$, $\displaystyle {a_{n+1}}={a_n+5}$ says that the sequence is arithmetic, with $\displaystyle {a_1}={4}$ and $\displaystyle {d}={5}$. The formula: $\displaystyle {a_n}={4+(n-1)\cdot5}={5n-1}$.
• In the formula $\displaystyle {a_n}={5n+3}$: $\displaystyle {a_{n+1}-a_n}={(5(n+1)+3)-(5n+3)}={5}$, and $\displaystyle {a_1}={8}$. The rule: $\displaystyle {a_1}={8}$, $\displaystyle {a_{n+1}}={a_n+5}$.

The arithmetic sequence: $\displaystyle 3, 10, 17, \ldots$

In which place is the term 346?

Is 100 a term of the sequence? Explain.

The arithmetic sequence: $\displaystyle 50, 47, 44, \ldots, -31$.

How many terms does the sequence have?

How many of its terms are positive?

What is the smallest positive term?

A sequence is defined by the recursion rule $\displaystyle {a_1}={-6}$, $\displaystyle {a_{n+1}}={a_n+4}$.
Write its general term, and simplify it.

In an arithmetic sequence $\displaystyle {a_1}={-20}$ and $\displaystyle {d}={3}$.
From which place are the terms positive?