$1.$ Following are the actual returns of the Dow Jones Industrial Average in the past $5$ years

Year DJIA return S&P $500$ return

$201725\mathrm{.}08\%19\mathrm{.}42\%$

$2018-5\mathrm{.}63\%-6\mathrm{.}24\%$

$201922\mathrm{.}34\%28\mathrm{.}88\%$

$20207\mathrm{.}25\%16\mathrm{.}26\%$

$202118\mathrm{.}73\%26\mathrm{.}89\%$

a$.$ Just by looking at the numbers, what are your expectations for the $5-$year arithmetic averages for the DJIA and the S&P $500?($i$.$e$.$ is there a range of values that you

b$.$ Calculate the $5-$year arithmetic average AND the $5-$year geometric average for the DJIA and the S&P $500.$

c$.$ Explain the numbers that you calculated in b for the DJIA in plain English. $($i$.$e$.$ if you calculated $10\%$ for the Dow$$s $5-$year arithmetic average, what does it mean in plain English? If you calculate $11\%$ for the Dow$$s $5-$year geometric average, what does it mean in plain English?$)$

d$.$ Comparing your numbers of arithmetic averages and geometric averages, what are your take away in terms of the relationship between the $2$ averages? $($is one type of average always greater than the other, and more importantly, WHY?$)$