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  • Find a, b and c such that the following numbers are in AP: a, 7, b, 23, c.

Find a, b and c such that the following numbers are in AP: a, 7, b, 23, c.

Answers (1)

Given AP is a, 7, b, 23, c

We know that if series are in AP then

a\textsubscript{1}= a, a\textsubscript{2}= 7, a\textsubscript{3}= b, a\textsubscript{4}=23, a\textsubscript{5}= c \\

(We know that the common difference of an A.P is Equal)

\\a\textsubscript{2} - a\textsubscript{1} = a\textsubscript{3} - a\textsubscript{2 }= a\textsubscript{4} - a\textsubscript{3} = a\textsubscript{5} - a\textsubscript{4}\\ \therefore 7 - a = b - 7 = 23 - b = c - 23\\ $Taking 2\textsuperscript{nd} and 3\textsuperscript{rd} term\\$ b - 7 = 23 - b\\ 2b = 30\\ b=\frac{30}{2}=15 \\ $Taking 1\textsuperscript{st} and 2\textsuperscript{nd} term\\$ 7 - a = b - 7\\ 7-a=15-7 ( \because b = 15)\\ 7 - 8 = a\\ -1 = a\\ $Taking 1\textsuperscript{st} and fourth term\\$ 7 - a = c - 23\\ 7-(-1)=c-23 (\because a = -1)\\ 7 + 1 = c - 23\\ 8 + 23 = 6\\ 31 = c \\

Hence a = - 1, b = 15, c = 31

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