1- An investment account offers an interest rate of 8% per annum compounded yearly.Assuming a single deposit £39700 is placed in this invest

2)  1000 pounds at the beginning of each half year

interest rate is 0.03 each half year

time period is 15 half years

Since payments are at the beginning of each period it is an annuity due

Since there is no money at the beginning and yourwant to know how much there is at the end it is a future value.

so

you need to use the future value of an annuity due formula with all the values I have given you.

Can you take it from there ?  :)

1)

At the end of year 1 he will have 39700*1.08-100 = 42776.00

At the end of year 2 he will have  42776*1.08-100=46098.08

then

$$\\68900=49098.08(1+0.08)^n\\\\
68900/49098.08=(1.08)^n\\\\
log(68900/49098.08)=log(1.08)^n\\\\
log(68900/49098.08)=nlog(1.08)\\\\
\frac{log(68900/49098.08)}{log1.08}=n\\\\
n=\frac{log(68900/49098.08)}{log1.08}\\\\$$

 

$${\frac{{log}_{10}\left({\frac{{\mathtt{68\,900}}}{{\mathtt{49\,098.08}}}}\right)}{{log}_{10}\left({\mathtt{1.08}}\right)}} = {\mathtt{4.402\: \!698\: \!337\: \!383\: \!920\: \!9}}$$

 

round up to 5 because he doestn't get the interest until the end of the year.

2+5=7 years