The sum of an arithmetic sequence

To work out $\displaystyle 1+2+3+\ldots+100$, we write the sum twice – once from the start and once from the end – and add them, number under number:

• Each column adds up to 101, and there are 100 columns: $\displaystyle {2S}={100\cdot101}$.
• $\displaystyle {S}={5050}$.

The same idea works in every arithmetic sequence: in the top row the terms go up by $\displaystyle d$, in the bottom row down by $\displaystyle d$, so each column has the same sum, $\displaystyle a_1+a_n$. There are $\displaystyle n$ columns:

Put in $\displaystyle {a_n}={a_1+(n-1)d}$, and we get a form without $\displaystyle a_n$:

$\displaystyle S_n$ is $\displaystyle n$ times the mean of the first and the last term.

The same proof in a picture: stairs of $\displaystyle 1+2+\ldots+n$ squares, and another copy of them upside down, make a rectangle of $\displaystyle n$ by $\displaystyle n+1$ squares. So $\displaystyle {1+2+\ldots+n}={\frac{n(n+1)}{2}}$.

The sums $\displaystyle S_1, S_2, S_3, \ldots$ are a sequence themselves – the sequence of partial sums. In the sequence $\displaystyle 1, 3, 5, \ldots$:

• $\displaystyle {S_n}={\frac{n\left[2+(n-1)\cdot2\right]}{2}}={n^2}$: the sums $\displaystyle 1, 4, 9, 16, \ldots$
• In every arithmetic sequence $\displaystyle {S_n}={\frac{d}{2}n^2+\left(a_1-\frac{d}{2}\right)n}$: a quadratic function of $\displaystyle n$ with no constant term (when $\displaystyle d\neq0$).
• So the points $\displaystyle (n, S_n)$ lie on a parabola through the origin:

How many terms of the sequence $\displaystyle 5, 8, 11, \ldots$ must be added to get 220?

• $\displaystyle {\frac{n\left[10+(n-1)\cdot3\right]}{2}}={220}$, that is $\displaystyle {n(3n+7)}={440}$, so $\displaystyle {3n^2+7n-440}={0}$.
• $\displaystyle {n}={\frac{-7\pm\sqrt{49+5280}}{6}}={\frac{-7\pm73}{6}}$: $\displaystyle {n}={11}$ or $\displaystyle {n}={-\frac{40}{3}}$.
• $\displaystyle n$ is natural, so 11 terms are needed. When the equation gives no natural $\displaystyle n$, no number of terms has exactly the given sum.

$\displaystyle S_n$ is $\displaystyle S_{n-1}$ plus the $\displaystyle n$th term. So the terms can be found from the sums:

and $\displaystyle {a_1}={S_1}$. If $\displaystyle {S_n}={2n^2+3n}$:

• $\displaystyle {a_n}={(2n^2+3n)-\left(2(n-1)^2+3(n-1)\right)}={4n+1}$, and $\displaystyle {a_1}={S_1}={5}$ fits the formula.
• $\displaystyle {a_{n+1}-a_n}={4}$: the sequence is arithmetic.

The sum: $\displaystyle 4+9+14+\ldots+99$.

How many terms has the sum?

Calculate the sum.

Calculate the sum of all two-digit numbers divisible by 3.

In an arithmetic sequence $\displaystyle {a_1}={-20}$ and $\displaystyle {d}={3}$.
From what number of terms is the sum $\displaystyle S_n$ positive? Solve an inequality, and write the answer.

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={3n^2-n}$.

Find $\displaystyle a_n$.

Prove that the sequence is arithmetic. (Check $\displaystyle a_1$ too.)

Is there a number of terms of the sequence $\displaystyle 5, 8, 11, \ldots$ whose sum is exactly 500? Solve an equation, and explain.