Question 815543
At the start, they deposit £5100.
In the first year the interest earned is the initial balance at the start of that year (£5100) times the interest rate as a fraction or decimal
{{{"( 4 % =" }}}{{{4/100=0.04}}} .
At the end of the first year, they will have the initial amount, plus the interest:5100
{{{"£ 5100 + £"}}}{{{5100*0.04= "£"}}}{{{(1+0.04)= "£"}}}{{{5100*1.04}}} .
That means that in the first year the initial balance grows by a factor of {{{1.04}}} .
That happens no matter what the initial balance, and no matter what happened before.
 At the beginning of the second year, they deposit £5100 again,
and end up with (in £)
{{{5100*1.04+5100}}}
Over the second year that amount of money again multiplies times {{{1.04}}} to get to
{{{(5100*1.04+5100)1.04=5100*1.04^2+5100*1.04}}}
by the end of the second year.
 
At the beginning of the third year, they deposit £5100 again,
and end up with (in £)
{{{5100*1.04^2+5100*1.04+5100}}}
Over the third year that amount of money again multiplies times {{{1.04}}} to get to
{{{(5100*1.04^2+5100*1.04+5100)1.04=5100*1.04^3+5100*1.04^2+5100*1.04}}}
by the end of the third year.
 
The pattern continues, and by the end of the fifth year,their balance is
{{{5100*1.04^5+5100*1.04^4+5100*1.04^3+5100*1.04^2+5100*1.04}}}
={{{5100*1.04*(1.04^4+1.04^3+1.04^2+1.04+1)}}}
={{{5100*1.04*(1.04^5-1)/(1.04-1)=5100*1.04*(1.04^5-1)/0.04}}}