Payments of $$5,000$ are made into a fund at the end of each year for

$20$ years. The fund receives interest at an annual effective rate of i$\%.$ The interest

generated is reinvested at an annual effective rate of j$\%.$ The money continues to

accumulate as described, even after deposits are terminated. Find a formula for the

total accumulated value at the end of twenty$-$five years using appropriate actuarial

symbols and rates i or j$.$

the answer is total $=5000$x$20+5000$i$($S$\_20$ at rate j$)(1+$j$)^4=20$X$5000$i $($S$\_4$ at rate j$)$