For S_n=a+ar+. s+arn-1, multiplying by r shifts the powers: rS_n=ar+ar2+. s+arn. Subtraction cancels the middle terms. (1-r)S_n=a-arn, so S_n= a(1-rn)1-r for r 1. Equivalently use a(rn-1)r-1. For r=1, all terms equal a, so S_n=na. For 3,6,12,, u_5=3(2)4=48 but S_5= 3(25-1)2-1=93. A deposit that increases by 5% each month has ratio 1.05; a quantity that decreases by 20% has ratio 0.8. Use a GP sum only when adding successive amounts, not when asking for the amount at one time. Keep full calculator precision until the final answer. State whether the initial event is included in a total.

For \(S_n=a+ar+\cdots+ar^{n-1}\), multiplying by \(r\) shifts the powers: \(rS_n=ar+ar^2+\cdots+ar^n\). Subtraction cancels the middle terms.

\((1-r)S_n=a-ar^n\), so \(S_n=\dfrac{a(1-r^n)}{1-r}\) for \(r\ne1\). Equivalently use \(\dfrac{a(r^n-1)}{r-1}\). For \(r=1\), all terms equal \(a\), so \(S_n=na\).

For \(3,6,12,\ldots\), \(u_5=3(2)^4=48\) but \(S_5=\dfrac{3(2^5-1)}{2-1}=93\).

A deposit that increases by \(5\%\) each month has ratio \(1.05\); a quantity that decreases by \(20\%\) has ratio \(0.8\). Use a GP sum only when adding successive amounts, not when asking for the amount at one time.

Keep full calculator precision until the final answer. State whether the initial event is included in a total.

Adding a geometric progression

Adding a geometric progression