A charity sets up a trust fund with an initial investment. It pledges to invest $0\mathrm{.}8m$ at the end of

each year. The fund is invested at an annual interest rate of $5\%$ which is added at the end of each

year.

$($i$)$ Find the first order difference equation for the value of the fund $(V_t)$ at the end of year $t.$

$($ii$)$ Find the general solution to the difference equation in terms of the initial investment, $$X$m.$ Use the

general solution to find the value of X which will ensure the fund is worth at least $15m$ at the end of the

${10}^{th}$ year.