JEE Main & Advanced Mathematics Sequence & Series Relation Between A.P., G.P. and H.P.

(1) If \[A,\,\,G,\,\,\,H\] be A.M., G.M., H.M. between \[a\] and \[b,\] then

\[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}=\left\{ \begin{align} & A\text{ when }n=0 \\ & G\text{ when }n=-1/2 \\ & H\text{ when }n=-1 \\\end{align} \right.\]

(2) If \[{{A}_{1}},\,{{A}_{2}}\] be two A.M.?s; \[{{G}_{1}},\,{{G}_{2}}\] be two G.M.?s and \[{{H}_{1}},\,{{H}_{2}}\] be two H.M.?s between two numbers \[a\] and \[b,\] then

\[\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\]

(3) Recognization of A.P., G.P., H.P. : If \[a,\,\,b,\,\,c\] are three successive terms of a sequence.

If  \[\frac{a-b}{b-c}=\frac{a}{a}\], then \[a,\,\,b,\,\,c\] are in A.P.

If, \[\frac{a-b}{b-c}=\frac{a}{b}\], then \[a,\,\,b,\,\,c\] are in G.P.

If, \[\frac{a-b}{b-c}=\frac{a}{c}\], then \[a,\,\,b,\,\,c\] are in H.P.

(4) If number of terms of any A.P./G.P./H.P. is odd, then A.M./G.M./H.M. of first and last terms is middle term of series.

(5) If number of terms of any A.P./G.P./H.P. is even, then A.M./G.M./H.M. of middle two terms is A.M./G.M./H.M. of first and last terms respectively.

(6) If \[{{p}^{th}},\,\,{{q}^{th}}\] and \[{{r}^{th}}\] terms of a G.P. are in G.P. Then \[p,\,\,q,\,\,r\] are in A.P.

(7) If \[a,\,\,b,\,\,c\] are in A.P. as well as in G.P. then \[a=b=c\].

(8) If \[a,\,\,b,\,\,c\] are in A.P., then \[{{x}^{a}},\,{{x}^{b}},\,{{x}^{c}}\] will be in G.P. \[(x\ne \pm 1)\].