1. Find Time n = ?
Regular Deposit
(PMT Amount) C = 1000, Interest Rate i = 10%, Future value FV = 6715.61,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Future value of Annuity Due method

Solution:
`C=1000` (Cash flow per year)

`i=10%=0.1` per year (Interest rate)

`FV=6715.61` (Future value)

Now, Future value (Annuity Due) formula is
`FV_("Annuity Due")=C*[((1+i)^n-1)/(i)]*(1+i)`

`:.6715.61=1000*[((1+0.1)^n-1)/(0.1)]*(1+0.1)`

`:.6715.61=1100*[((1.1)^n-1)/(0.1)]`

`:.(6715.61)/(1100)*0.1=(1.1)^n-1`

`:.0.61=(1.1)^n-1`

`:.1.1^n=1+0.61`

`:.1.1^n=1.61`

taking natural log on both the sides
`:.ln(1.1^n)=ln(1.61)`

`:.n*ln(1.1)=ln(1.61)`

`:.n=ln(1.61)/ln(1.1)`

`:.n=(0.48)/(0.1)`

`:.n=5` year

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2. Find Time n = ?
Regular Deposit
(PMT Amount) C = 5000, Interest Rate i = 10%, Future value FV = 18205,
Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year)
for Future value of Annuity Due method

Solution:
`C=5000` (Cash flow per year)

`i=10%=0.1` per year (Interest rate)

`FV=18205` (Future value)

Now, Future value (Annuity Due) formula is
`FV_("Annuity Due")=C*[((1+i)^n-1)/(i)]*(1+i)`

`:.18205=5000*[((1+0.1)^n-1)/(0.1)]*(1+0.1)`

`:.18205=5500*[((1.1)^n-1)/(0.1)]`

`:.(18205)/(5500)*0.1=(1.1)^n-1`

`:.0.33=(1.1)^n-1`

`:.1.1^n=1+0.33`

`:.1.1^n=1.33`

taking natural log on both the sides
`:.ln(1.1^n)=ln(1.33)`

`:.n*ln(1.1)=ln(1.33)`

`:.n=ln(1.33)/ln(1.1)`

`:.n=(0.29)/(0.1)`

`:.n=3` year

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