Suppose you make a deposit of $P into a savings account that earns interest at a

Question

Suppose you make a deposit of $P into a savings account that earns interest at a rate of 100r % per year. Show that if interest is compounded once per year, then the balance after t years is B(t) = P(1 + r)^t. If interest is compounded m times per year, then the balance after t years is B(t) = P(1 + r/m)^mt. For example, m = 12 corresponds to monthly compounding, and the interest rate for each month is r/12. In the limit m rightarrow infinity, the compounding is said to be continuous. Show that with continuous compounding, the balance after t years is B(t) = Pe^rt.

Explanation / Answer

B(t)==p(1+r)t

a)balance after 1 year =p +pr =p(1+r)

balance after 2 years =p(1+r) +p(1+r)r =p(1+r)(1+r)=p(1+r)2

let balance after t years B(t)=p(1+r)t is true

after t+1 years  B(t+1)=p(1+r)t + rp(1+r)t

B(t+1)=p(1+r)t (1+ r)

B(t+1)=p(1+r)t+1 satisfied

hence proved by induction

b)B(t)=p(1+ r/m)mt

when compounded continously

B(t)=limm->p(1+ r/m)mt

B(t) /p=limm->(1+ r/m)mt

apply logarithm on both sides

ln((B(t))/p)=limm-> ln (1+ r/m)mt

ln((B(t))/p)=limm-> mt(ln (1+ r/m))

ln((B(t))/p)=limm-> t(ln(1+ r/m)) /(1/m)

apply l hospitals rule differentiate numerator,denominator with respect to m

ln((B(t))/p)=limm-> t((1/(1+ r/m))*(-r/m2)) /(-1/m2)

ln((B(t))/p)=limm-> t((1/(1+ r/m))*(r))

ln((B(t))/p)=  t((1/(1+0))*(r))

ln((B(t))/p)=  tr

((B(t))/p)=etr

B(t)=petr

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.