I noticed there did not necessarily seem to be a caveat for adjusting contribution frequency. I have included a formula below that would take this into account.
A = P(1+r/n)^(nt) + c[a(1 - r/n)^(nfz)] / [1 - (1 + r/n)^(nf)]
P = Principal
r = interest rate
n = number of compounds per year
t = number of years this is compounding
c = the amount of the contributions made each period
a = will be one of two things depending on when contributions are made [if made at the end of the period, a = 1. If made at the beginning of the period, a = (1 + r/n)^(n*f)]
f = frequency of contributions in years (so if monthly, f = 1/12)
z = the number of contributions you would make over the life of the account (typically this would be t/f)
For example, suppose I had $10,000 in an account compounding daily at 4%. If I make contributions monthly of $100, then what is the value in 10 years? This would be set up accordingly.
Contributions made at the end of the month:
A = 10,000(1 + 0.04/365)^(365 * 10) + 100[1(1 - 0.04/365)^(365 1/12(10/(1/12))] / [1 - (1 + 0.04/365)^(365*1/12)]
Simplifying:
A = 10,000(1 + 0.04/365)^(3,650) + 100[1(1 - 0.04/365)^(3,650)] / [1 - (1 + 0.04/365)^(365/12)]
A = $29,647.91
Contributions made at the beginning of the month:
A = 10,000(1 + 0.04/365)^(365 * 10) + 100[(1 + 0.04/365)^(365*1/12)(1 - 0.04/365)^(365 1/12(10/(1/12))] / [1 - (1 + 0.04/365)^(365*1/12)]
Simplifying:
A = 10,000(1 + 0.04/365)^(3,650) + 100[(1 + 0.04/365)^(365/12)(1 - 0.04/365)^(3,650)] / [1 - (1 + 0.04/365)^(365/12)]
A = $29,697.09