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How do I find the sum of an arithmetic sequence on a calculator?

Answer 1

Anna Allen

You can use the following general formula:

#sum_(k=1)^n(ak+b)=n/2[a(n+1)+2b]#

where the series has #n# terms #a# and #b# are constants.

(Ask a general question, get a general answer.)

Seems that it would be handy to have a closed form expression for the following:

#sum_(k=1)^n(ak+b)#

where a and b are constants.

The commutative property of addition says

#sum_(k=1)^n(ak+b)=sum_(k=1)^nak+sum_(k=1)^nb#

We can factor the #a#.

#sum_(k=1)^n(ak+b)=asum_(k=1)^nk+sum_(k=1)^nb#

But we know closed form representations for both of these sums.

#sum_(k=1)^nk=(n(n+1))/2#

and

#sum_(k=1)^nb=nb#

#sum_(k=1)^n(ak+b)=(an(n+1))/2+nb=n/2[a(n+1)+2b]#

EXAMPLE

Suppose we wanted the following sum:

#7+10+13+16+19+22+25+28+31#

There are 9 terms in this series so #n=9#. Consecutive terms differ by 3 so #a=3#. The first term, #x_1=7# and so #b=x_1-a=7-3=4#.

We want to find

#sum_(k=1)^9(3k+4)=9/2[3(9+1)+2*4]=171#

which is, in fact, the sum of the numbers given.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.