Question

Look at the attached picture & please help!

If that pattern continues, how many squares will be in the 50th term and how many of these squares will be shaded?

Answer 1

I think you need to provide an image of the pattern

Answer 2

sorry! @campbell_st can you see it now?

Answer 3

so far for my equation I have: 2n because each stage is being multiplied by 2. am i on the right track?

Answer 4

@Hero @mathmath333 @MrNood

Answer 5

\(\large \color{black}{\begin{align} 2(1)-1\hspace{.33em}\\~\\
2(1+2)-1\hspace{.33em}\\~\\
2(1+2+3)-1\hspace{.33em}\\~\\
2(1+2+3+4)-1\hspace{.33em}\\~\\
\cdots
\implies
2\color{red}{(1)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2+3)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2+3+4)}-1\hspace{.33em}\\~\\
\cdots \hspace{.33em}\\~\\


\end{align}}\)

now if u observe the red numbers they are represented as triangular numbers

which as the formula of \(\large n^{th}\) term as \(\large \dfrac{n(n+1)}{2}\)

so the fomula for your \(\large n^{th}\) term is


\(\large \color{black}{\begin{align}
2 \dfrac{n(n+1)}{2}-1
\end{align}}\)

substitute \(50\) in place of \(n\) and get the answer

Answer 6

the number of total squares are if u count...
\(\large \color{black}{\begin{align} 1\hspace{.33em}\\~\\
5\hspace{.33em}\\~\\
11\hspace{.33em}\\~\\
19\hspace{.33em}\\~\\
\cdots


\end{align}}\)


they can be written as


\(\large \color{black}{\begin{align} 2(1)-1\hspace{.33em}\\~\\
2(1+2)-1\hspace{.33em}\\~\\
2(1+2+3)-1\hspace{.33em}\\~\\
2(1+2+3+4)-1\hspace{.33em}\\~\\
\cdots
\implies
2\color{red}{(1)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2+3)}-1\hspace{.33em}\\~\\
2\color{red}{(1+2+3+4)}-1\hspace{.33em}\\~\\
\cdots \hspace{.33em}\\~\\


\end{align}}\)

now if u observe the red numbers they are represented as triangular numbers

which as the formula of \(\large n^{th}\) term as \(\large \dfrac{n(n+1)}{2}\)

so the fomula for your \(\large n^{th}\) term is


\(\large \color{black}{\begin{align}
2 \dfrac{n(n+1)}{2}-1
\end{align}}\)

substitute \(50\) in place of \(n\) and get the answer

Answer 7

would the answer be 2550?

Answer 8

how do i know how many squares would be shaded?

Answer 9

i m getting \(2549\)

Answer 10

hm..

Answer 11

for the second part , i am getting a floor function , if u dont know about that take a look here
, https://www.mathsisfun.com/sets/function-floor-ceiling.html


\(\large 1,3,7,11,17,23\cdots\)

shaded terms =\(\Large \lfloor{\dfrac{(n+1)^2}{2}}\rfloor-1\)

Answer 12

by plugging \(50\) u will get the answer of shaded squares

Answer 13

https://oeis.org/search?q=1%2C3%2C7%2C11%2C17...&sort=&language=english&go=Search

Answer 14

it should be \(1229\)