b) If [tex]$\frac{l+a}{n-1}= d$[/tex] and [tex]$2 s= n ( a + l )$[/tex], express s in terms n and d only.
Answer (1)
Express l + a in terms of d and n : l + a = d ( n − 1 ) .
Substitute this expression into the second equation: 2 s = n ( d ( n − 1 )) .
Solve for s : s = 2 n d ( n − 1 ) .
Express s in terms of n and d : s = 2 n ( n − 1 ) d .
Explanation
Understanding the Problem We are given two equations: n − 1 l + a = d 2 s = n ( a + l ) Our goal is to express s in terms of n and d only.
Expressing l+a From the first equation, we can express l + a in terms of d and n :
l + a = d ( n − 1 )
Substitution Substitute this expression into the second equation: 2 s = n ( d ( n − 1 ))
Solving for s Now, solve for s :
s = 2 n d ( n − 1 ) s = 2 1 n ( n − 1 ) d So, s is expressed in terms of n and d .
Final Answer Therefore, s in terms of n and d is: s = 2 n ( n − 1 ) d
Examples
Consider a scenario where you are arranging a series of equally spaced items. If 'd' represents the constant difference between the positions of these items, and 'n' is the number of items, then 's' could represent a measure related to the total spacing or arrangement complexity. For instance, if you are designing a display or organizing elements in a pattern, understanding how 's' changes with 'n' and 'd' helps in optimizing the layout and predicting the overall structure. This formula allows you to quickly calculate 's' based on the number of items and the constant difference, aiding in efficient planning and design.
