<prompt> Fumana equation bakeng sa ho bala phaello e kopaneng.nan  <response>: Instruction in English: Derive the equation for calculating compound interest. . Response in English: Compound interest is the interest earned not only on the initial principal amount invested but also on the accumulated interest of previous periods. The formula used to calculate compound interest is derived from the basic concept of exponential growth.Lets consider an initial principal amount, P, that is invested at an annual interest rate, r, compounded n times per year, for a total time of t years. The interest earned after the first period will be Prn, and the new principal amount becomes P Prn.After the second period, the interest earned is P Prnrn, and the new principal amount becomes P Prn P Prnrn.We can simplify this expression by factoring out P to get P1 rn P1 rnrn P1 rn2.Continuing the process, after the third period, the new principal amount becomes P1 rn3. After the fourth period, it is P1 rn4, and so on.Thus, after t years, the total number of periods is nt, and the final amount, A, can be calculated as:A P1 rnntWhere,A final amountP initial principal amountr annual interest rate as a decimaln number of compounding periods per yeart total time in yearsThis is the formula for calculating compound interest. . Response in Sesotho: Tsoala e kopaneng ke tsoala e fumanoeng eseng feela cheleteng ea mantlha e tsetelitsoeng empa hape le phaellong e bokelletsoeng ea linako tse fetileng. Mokhoa o sebelisoang ho bala phaello e kopaneng e tsoa mohopolong oa mantlha oa kholo ea kholo. Ha re nahaneng ka chelete ea mantlha ea mantlha, P, e tseteletsoeng ka sekhahla sa tsoala ea selemo, r, e kopantsoeng ka makhetlo a n ka selemo, bakeng sa nako eohle ea lilemo tse t. Tsoala e tla fumanoa ka mora nako ea pele e tla ba Prn, me chelete e ncha e tla fetoha P Prn. Ka mora nako ea bobeli, phaello e fumanoeng ke P Prn rn, me chelete e ncha ea sehlooho e fetoha P Prn P Prn rn. Re ka nolofatsa polelo ena ka ho beha P ho fumana P1 rn P1 rnrn P1 rn2 . Ho tsoela pele ka tsebetso, ka mora nako ea boraro, chelete e ncha ea sehlooho e fetoha P1 rn3. Ka mora nako ea bone, ke P 1 rn4, joalo-joalo. Kahoo, ka mora lilemo tse t, kakaretso ea linako ke nt, me chelete ea ho qetela, A, e ka baloa e le: A P1 rnnt Moo, A palo ea ho qetela P chelete ea mantlha ea mantlha r sekhahla sa tsoala sa selemo e le desimali n palo ea linako tse kopanyang ka selemo t kakaretso ea nako ea lilemo Ena ke mokhoa oa ho bala phaello e kopaneng. 