Anonymous

# In 1971 a newly-built flat was sold with a 999-year lease.?

The terms of the sale included a requirement to pay groun rent yearly. The ground rent was set at \$28 per year for the first 21 years of the lease, increasing by \$14 to \$42 per year for the next 21 years, and then increasing again by \$14 at the end of each subsequent period of 21 years.

(a)Find how many complete 21-yer periods there would be if the lease ran for the full 999 years, and how many years there would be left over.

(b)Find the total amount of ground rent that would be paid in all of the complete 21-year periods of the lease.

Relevance

999 / 21 = 47.57..

21 * 47 = 987

999 - 987 = 12

a) 47 full sets of 21 years, with an additional 12 years at the end

The amount of rent per 21 year period is:

21*(28+N*14) , where N is the Nth 21 year period, starting at 0.

Calculating the total amount for the 999 year lease is straightforward:

21*28 + 21*(28+14) + 21*(28+2*14) ... 21*(28+46*14) + 12*(28+47*14)

21*( 28 + (28+14) + (28+2*14) + ... (28+ 46*14)) + 12*(28+47*14)

21*( 47*28 + 14 + 2*14 + ... 46*14) + 12*(28+47*14)

Theres the trick here, where we simplify the 1 + 2 + 3 + ... + N down to (N^2 + N) / 2

21*( 47*28 + ((46^2+46)/2)*14 ) + 12*(28+47*14)

21*( 47*28 + 1081*14 ) + 12*(28+47*14)

21*( 1316 + 15134 ) + 12*( 28 + 658 )

21*16450 + 12*18424

345450 + 221088 = \$565,950 total ground cost for the full 999 years, \$345,450 for the 47 full 21 year periods, and \$221,088 for the last 12 years.

Edit - Oops, I had an off-by-1 error that I have corrected. :)

• a) 999/21 = 47.57 => there are 47 complete 21 year periods

b) Let k represent the kth 21-year period.

k=1: rent is 28 per year

k=2: rent is 28+14 per year

k=3: rent is 28+14+14 per year

...

k=47: rent is 28+14*(k-1) per year

Total rent would be Sum[21*(28+14*(k-1))] as k ranges from 1 to 47 = Sum[588+294(k-1)] as k ranges from 1 to 47.

The sum of an arithmetic series is (n/2)*(first term + last term) where n = 47 (in this problem).

Total rent in 47 complete 21-year periods of the lease

=(47/2)*(588+14112)=\$345,450

• Anonymous