Finding the nth line of a sequence. Help with the logic

[MathsFormula]

The sequence is:

LINE 1 1/2 +1/4 = 3/4

LINE 2 1/2 +1/4 + 1/8 = 7/8

LINE 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16)

LINE 3 1/2 +1/4 + 1/8 + 1/32 = ?? (Answer is 31/32)

Q1. Fill in the missing ?? answers.

Q2. Write the answer to the nth line of the sequence



I managed to calculate the missing ?? answers okay but could not get Q2

The answer in the book is 1- (1/2ⁿ⁺¹)

I don't see how someone can work through the problem logically, step by step and get an answer so complicated. Seems like either the person can just 'see it' or not.

Please can some one guide me step by step to getting the answer. I don't know where to start.

 

[Deleted member 4993]

The sequence is:

LINE 1 1/2 +1/4 = 3/4 → numerator = 2^(1+1)-1 & denominator = 2^(1+1)

LINE 2 1/2 +1/4 + 1/8 = 7/8 → numerator = 2^(2+1)-1 & denominator = 2^(2+1)

LINE 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16) → numerator = 2^(3+1)-1 & denominator = 2^(3+1)

LINE 3 1/2 +1/4 + 1/8 + 1/32 = ?? (Answer is 31/32) → numerator = 2^(4+1)-1 & denominator = 2^(4+1)

Q1. Fill in the missing ?? answers.

Q2. Write the answer to the nth line of the sequence



I managed to calculate the missing ?? answers okay but could not get Q2

The answer in the book is 1- (1/2ⁿ⁺¹)

I don't see how someone can work through the problem logically, step by step and get an answer so complicated. Seems like either the person can just 'see it' or not.

Please can some one guide me step by step to getting the answer. I don't know where to start.

.

 

[pka]

The sequence is:

LINE 1 1/2 +1/4 = 3/4

LINE 2 1/2 +1/4 + 1/8 = 7/8

LINE 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16)

LINE 3 1/2 +1/4 + 1/8 + 1/32 = ?? (Answer is 31/32)
Q1. Fill in the missing ?? answers.
Q2. Write the answer to the nth line of the sequence

Your LINE 3 is not correct \(\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+ \frac {1}{32}=
\frac {31}{32}\)

\(\displaystyle \begin{align*} {S_n} &= r + {r^2} + \cdots {r^n} = \sum\limits_{k = 1}^n {{r^k}} \\ r{S_n} &= {r^2} + {r^3} + \cdots {r^{n + 1}} = \sum\limits_{k = 1}^n {{r^{k + 1}}} \\ (1 - r){S_n} &= r - {r^{n + 1}} \\ {S_n} &= \sum\limits_{k = 1}^n {{r^k}} = \frac{{r - {r^{n + 1}}}}{{1 - r}}\end{align*}\)

I think the answer given by your text is mistaken.
\(\displaystyle \dfrac{{\frac{1}{2} - \frac{1}{{{2^{5 + 1}}}}}}{{1 - \frac{1}{2}}} = ?\)

 

[Ishuda]

The sequence is:

LINE 1 1/2 +1/4 = 3/4

LINE 2 1/2 +1/4 + 1/8 = 7/8

LINE 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16)

LINE 3 1/2 +1/4 + 1/8 + 1/32 = ?? (Answer is 31/32)

Q1. Fill in the missing ?? answers.

Q2. Write the answer to the nth line of the sequence



I managed to calculate the missing ?? answers okay but could not get Q2

The answer in the book is 1- (1/2ⁿ⁺¹)

I don't see how someone can work through the problem logically, step by step and get an answer so complicated. Seems like either the person can just 'see it' or not.

Please can some one guide me step by step to getting the answer. I don't know where to start.

Several ways to go about this, some easier than others. For example, Letting L_n indicate the value for line n, we might note that
L_n+1 = (1/2) + (1/2) L_n
or
LINE 1 1/2 +1/4 = 3/4 = 1 - 1/4

LINE 2 1/2 +1/4 + 1/8 = 7/8 = 1 - 1/8

LINE 3 1/2 +1/4 + 1/8 + 1/16 = 1 - 1/16
...
Noting the last number for LINE n is 1/2ⁿ⁺¹ gives the answer for LINE n.

Also one could note
L_n = 1/2+1/4+1/8+1/16+...+1/2ⁿ⁺¹ = (1/2) [1+1/2+1/4+1/8+1/16+...+1/2ⁿ]
and use the geometric series addition formula.

 

[MathsFormula]

SORRY I MADE A MISTAKE COPYING THE QUESTION FROM THE BOOK

The sequence is:

Line 1 1/2 +1/4 = 3/4

Line 2 1/2 +1/4 + 1/8 = 7/8

Line 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16)

Line 4 1/2 +1/4 + 1/8 + 1/16 + 1/32 = ?? (Answer is 31/32)


Q2. Write the answer to the nth line of the sequence.

The answer in the book is 1-(1/2ⁿ⁺¹)




I'm having trouble understanding your thought processes in getting to the answer (made worse by the fact the I wrote the question in correctly)

Here's my thinking :

n is the line number so to get the answer 3/4 for Line n = 1 the formula (n+2)/(n+3) applies


To get the answer 7/8 for Line n = 2 the formula (n+5)/(n+6) applies

To get the answer 15/16 for Line n = 3 the formula (n+12)/(n+13) applies

To get the answer 31/32 for Line n = 4 the formula (n+27)/(n+28) applies

Now I don't know what to do. I can't see a pattern and to just see 1-(1/2ⁿ⁺¹) in your mind is just a gift someone has. I don't have that gift so need to know how to get there step by step

 

[Harry_the_cat]

SORRY I MADE A MISTAKE COPYING THE QUESTION FROM THE BOOK

The sequence is:

Line 1 1/2 +1/4 = 3/4

Line 2 1/2 +1/4 + 1/8 = 7/8

Line 3 1/2 +1/4 + 1/8 + 1/16 = ?? (Answer is 15/16)

Line 4 1/2 +1/4 + 1/8 + 1/16 + 1/32 = ?? (Answer is 31/32)


Q2. Write the answer to the nth line of the sequence.

The answer in the book is 1-(1/2ⁿ⁺¹)




I'm having trouble understanding your thought processes in getting to the answer (made worse by the fact the I wrote the question in correctly)

Here's my thinking :

n is the line number so to get the answer 3/4 for Line n = 1 the formula (n+2)/(n+3) applies


To get the answer 7/8 for Line n = 2 the formula (n+5)/(n+6) applies

To get the answer 15/16 for Line n = 3 the formula (n+12)/(n+13) applies

To get the answer 31/32 for Line n = 4 the formula (n+27)/(n+28) applies

Now I don't know what to do. I can't see a pattern and to just see 1-(1/2ⁿ⁺¹) in your mind is just a gift someone has. I don't have that gift so need to know how to get there step by step

What you've said above is correct, but you are looking for ONE formula which works every time.

Notice that the denominators in the answer double each time ie \(\displaystyle 4, 8, 16, 32 \)etc, ie \(\displaystyle 2^2, 2^3, 2^4, 2^5 \).

Now Line 1 goes with \(\displaystyle 2^2\) or \(\displaystyle 2^{1+1}\).

Line 2 goes with \(\displaystyle 2^3\) or \(\displaystyle 2^{2+1}\) etc.

So Line n goes with \(\displaystyle 2^{n+1}\).

The numerator is always 1 less than the denominator, so it is \(\displaystyle 2^{n+1}-1\) is the numerator.

A bit of algebra tidies it up to get 1-(1/2ⁿ⁺¹).