Question Number 82540 by zainal tanjung last updated on 22/Feb/20
$$\mathrm{The}\:\mathrm{arithmetic}\:\mathrm{mean}\:\mathrm{between}\:\mathrm{two}\:\mathrm{numbers} \\ $$$$\mathrm{is}\:{A},\:\mathrm{and}\:{S}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:{n}\:\mathrm{arithmetic}\:\mathrm{means} \\ $$$$\mathrm{between}\:\mathrm{these}\:\mathrm{numbers},\:\mathrm{then} \\ $$
Answered by TANMAY PANACEA last updated on 22/Feb/20
![let two number is p,q given p,A,q [are in A.P] soA−p=q−A→2A=p+q.....(1) S is sum of n arith matic mean first term=p+d last term=q−d S=(n/2)[p+d+q−d] S=(n/2)×2A S=nA](Q82548.png)
$${let}\:{two}\:{number}\:{is}\:{p},{q} \\ $$$${given}\:{p},{A},{q}\:\:\left[{are}\:{in}\:{A}.{P}\right] \\ $$$${soA}−{p}={q}−{A}\rightarrow\mathrm{2}{A}={p}+{q}.....\left(\mathrm{1}\right) \\ $$$${S}\:{is}\:{sum}\:{of}\:\:{n}\:{arith}\:{matic}\:{mean} \\ $$$${first}\:{term}={p}+{d} \\ $$$${last}\:{term}={q}−{d} \\ $$$${S}=\frac{{n}}{\mathrm{2}}\left[{p}+{d}+{q}−{d}\right] \\ $$$${S}=\frac{{n}}{\mathrm{2}}×\mathrm{2}{A} \\ $$$${S}={nA} \\ $$$$ \\ $$$$\:\:\:\: \\ $$$$ \\ $$$$ \\ $$