The common difference of an arithmetic progression can be determined by given the first term and a subsequent term. In this case, the first term is 10 and the secondterm is four times the fifth term. To calculate the common difference, we must utilize the formula d = (a₂ – a₁) / (n₂ – n₁), where “a” is any two terms of the sequence and “n” are their respective positions in said sequence. Using this formula, we find that d = 40/4 = 10.

We can now use this value to determine the sum ofthefirst 12 terms usingtheformula Sᵣ = n⁄₂(2a + (n – 1)d). Plugging our values into this equation gives us S12=12/2(20+11*10)=780. Therefore, we conclude that when an arithmetic progression has a first term equal to 10 and its secondtermfourtimesthefifthterm,thencommon differenceis10andthesumofthefirst12terms equals 780.

These results should be kept in mind when considering various problems related tothis topic as they help provide general insight into how certain equations work solving them efficiently with minimal effort required complete task at hand! Furthermore understanding relationship between input variables output also key factor achieving maximum accuracy desired outcome order guarantee all objectives have been met expectations exceeded beyond initial scope work involved….

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