Further Pure 1 help

https://jacktilson.net/edu/maths/markschemes/summer-2005.pdf
If you go to page 25 and look at Question 5 , it doesnt make sense to me?
Can someone help?
Basically in the first line it says 2^(k) + 2^(k) -1 = 2^(k+1) -1 and then says 2.2^(k)= 2^(k+1)
Can someone explain it to me please?

https://jacktilson.net/edu/maths/markschemes/summer-2005.pdf
If you go to page 25 and look at Question 5 , it doesnt make sense to me?
Can someone help?
Basically in the first line it says 2^(k) + 2^(k) -1 = 2^(k+1) -1 and then says 2.2^(k)= 2^(k+1)
Can someone explain it to me please?

If we say a= 2^k

Then we have a + a -1. This becomes 2a - 1.

Put a back in to give 2* (2^k) + 1.

Then use what you know to get the 2^(k+1) from this statement.

Is this the part that needed clarifying?

Basic GCSE indices
$2\cdot 2^k = 2^1 \cdot 2^k = 2^{(1+k)}$

Its the 2^k +2^k that is not making sense tbh , like i know the part you said , that makes sense to me , its the top part how 2^k + 2^k= 2^(k+1)

For any number, call it x, we have x+x = 2x.

So, here $2^k +2^k = 2\cdot 2^k$

Then, as RDKGames said.

If we say a= 2^k

Then we have a + a -1. This becomes 2a - 1.

Put a back in to give 2* (2^k) + 1.

Then use what you know to get the 2^(k+1) from this statement.

Is this the part that needed clarifying?

Okay but how do we know those two equal each other?
Like I expect 2^k + 2^k to be something else? Sorry i am really stuck

Okay but how do we know those two equal each other?
Like I expect 2^k + 2^k to be something else? Sorry i am really stuck

What do you mean? You are adding $2^k$ to itself once so you have 2 lots of $2^k$

Its like saying $1+1=2 \cdot 1 = 2$ or $2+2+2=3\cdot 2 = 6$

Perhaps a specific example will help.

$2^5+2^5= 2\times 2^5 = 2^1\times 2^5=2^{5+1}=2^6$

Or putting it another way

$2^5+2^5=32+32=64=2^6=2^{5+1}$