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# Insert k arithmetic means between 1 and $k^2$ . what is the resulting series?

$(a)\;1,k,k^2\qquad(b)\;1,k,k+1,k+2\;---\;k^2\qquad(c)\;1,k,2k-1,3k-2\;---k^2-k+1,k^2\qquad(d)\;1,k,2k,3k\;--\;k^2$

Answer : (c) $1,k,2k-1,3k-2\;-------\;k^2-k+1,k^2$
Explanation : k arithmetic means between $\;1\;\xi\;k^2$
$t_{k+2}=k^2\quad\;,t_{1}=1$
$k^2=1+(k+1)d$
$d=\frac{k^2-1}{k+1}=k-1$
$Series\;is\;1,k,2k-1,3k-2\;--------\;k^2-k+1,k^2.$