A.B. deposits m*$100 into a bank account. The bank credits interest at nominal annual rate j, convertible semiannually, for the first seven

 

 I have left out M because down the bottom it says M=1

I have not included all of the initial working, but most is here. 

 

After 5 years            

 $$X=100(1+0.5J)^{10}\qquad(1)$$

 

After 7 years 

 

$$FV=100(1+0.5J)^{14}$$

 

After 10.5 years    (3.5 years more)

 

$$FV=[100(1.05J)^{14}](1+0.5J)^{14}=100(1.05J)^{28}$$

 

$$\\1.98*100=100(1+0.5J)^{28}\\\\
1.98=(1+0.5J)^{28}\\\\
(1.98)^{(1/28)}=1+0.5J\\\\
(1.98)^{(1/28)}-1=0.5J\\\\
2[(1.98)^{(1/28)}]-2=J\\\\
J=2[(1.98)^{(1/28)}]-2 \qquad(2)\\\\
$Sub into (1)$\\\\$$

 

$$\\X=100(1+0.5J)^{10}\qquad(1)\\\\
X=100(1+0.5*(2[(1.98)^{(1/28)}]-2)\\\\
X=100(1+(1.98)^{(1/28)}-1)\\\\
X=100(1.98)^{(1/28)}\\\\$$

 

$${\mathtt{100}}{\mathtt{\,\times\,}}\left({{\mathtt{1.98}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{28}}}}\right)}\right) = {\mathtt{102.469\: \!634\: \!086\: \!312\: \!44}}$$    

 

X=$102.47

 

I did the maths while I was writing tht LaTex so there could easily be mistakes.

I have not checked it.