Question

Summation problem

Answer 1

\[\sum_{k=1}^{45}(5k-1)\] What is the shortcut?

Answer 2

figure out it's partial sum and then plug in 45. to figure out it's partial sum, find a pattern from it's sum.

Answer 3

plug in the first view values and see what happens.

Answer 4

it's plus 5

Answer 5

now what..

Answer 6

first is 4 for k=1 then it's 9 for k=2 then it's 14 for k = 3

Answer 7

The fact that it goes up by 5 every time makes in an arithmetic sequence. To get the sum of the first n terms of an arithmetic sequence, do
\(\Large S_n = \frac{n}{2}(a_1+a_n)\)

In this case, it's the sum of the first 45, so n=45.

Answer 8

so it's (45/2)(5 + 225)?

Is this "Sn = n/2 (a1 + an)" true for all summation problems?

Answer 9

you can also try \[\sum_{k=1}^{45}(5k-1)=5\sum_{k-1}^{45}k-\sum_{k=1}^{45}1\]
\[=\frac{5\times 45\times 46}{2}-45\]

Answer 10

It's not true for all summation problems. You first have to look at the thing you are summing first and make sure that it's an arithmetic sequence.

Answer 11

why did I get 45 extra

Answer 12

maybe you forgot to subtract 45?

Answer 13

so it's (45/2)(5 + 225)?
\(\Large a_{45} = 224\)

Answer 14

45*5-1 = 224, not 225

Answer 15

oh i see now. So if you find that there is a sequence, you don't plug it in anywhere. The fact that a sequence exists allows us to use that formula?

Answer 16

Exactly. The arithmetic sequence has a special formula. If it's a different kind of sequence, there might be a different formula, or there may be no formula at all.