Let's consider only two operations: INSERT, in which a single element is inserted into an array with available space, and INSERTEXPAND, in which the array is full, so in order to insert an element we first copy all elements of the array into a new array twice the size $($doubling the array$).$

$($a$)$ First, use the accounting approach to prove that inserting $\backslash ($ n $\backslash )$ elements in a dynamic array needs $\backslash ($ O$($n$)\backslash )$ amortized time.

$($b$)$ Suppose we change our model for how much we increase the size of our array when it is full. For each of the three options below, indicate whether or not we can still achieve $\backslash ($ O$($n$)\backslash )$ amortized time for $\backslash ($ n $\backslash )$ operations. If yes, what would change in the previous proof argument? If not, explain why not.

i$.$ Quadrupling the array.

ii$.$ Increasing the size of the array by only $\backslash (10\backslash \%\backslash ).$

iii. Increasing the size of the array by a constant $10($each time$).$