Word Problem

[elmer]

Each row of *'s has two more *'s than the row immediately above it, as shown. Altogether, how many *'s are contained in the first twenty rows?
*
***
*****
*******
********* etc... to 21 rows

I can get the answer by writing out all the numbers from each row and adding them. Is there a short cut with a formula using slope and x,y coordinates?

 

[Deleted member 4993]

Each row of *'s has two more *'s than the row immediately above it, as shown. Altogether, how many *'s are contained in the first twenty rows?

I can get the answer by writing out all the numbers from each row and adding them. Is there a short cut with a formula using slope and x,y coordinates?

Where is it shown?

Have you done arithmatic progression in algebra?

 

[mmm4444bot]

as shown

*
***
*****
*******
*********
etc ... to 21 rows

I think that Subhotosh somehow missed the meaning of your diagram.



how many [asterisks] are contained in the first twenty rows ?

If you want the sum of the asterisks comprising the first 20 rows, why does your diagram describe 21 rows ?


Is there a short cut with a formula YES

using slope and x,y coordinates ? NO (not for pre-algebra)

This exercise is equivalent to asking, "What is the sum of the first 20 odd numbers ?"

Click HERE for Google results. You can read about the formula from many viewpoints.

Feel free to post additional specific questions here.

Cheers ~ Mark

 

[soroban]

Hello, elmer!

Each row of \(\displaystyle \circ\)'s has two more than the row immediately above it.

Altogether, how many \(\displaystyle \circ\)'s are contained in the first twenty rows?

. . \(\displaystyle \begin{array}{c} \circ \\ \circ\,\circ\,\circ \\ \circ\,\circ\,\circ\,\circ\,\circ \\ \circ\,\circ\,\circ\,\circ\,\circ\,\circ\,\circ \\ \hdots\hdots\hdots\hdots\hdots\hdots \end{array}\)

Did you try to count them?

. . \(\displaystyle \begin{array}{ccc}\text{Rows} & \text{Sum} & \text{Total} \\ \hline 1 & 1 & 1 \\ 1,2 & 1+3 & 4 \\ 1,2,3 & 1+3+5 & 9 \\ 1,2,3,4 & 1+3+5+7 & 16 \\ 1,2,3,4,5 & 1+3+5+7+9 & 25 \\ \vdots & \vdots & \vdots \end{array}\)
. . . . . . . . . . . . . . . . . . . . . . . . . \(\displaystyle \uparrow\)
. . . . . . . . . . . . . . . . . . . Do you see a pattern?