JEE Main & Advanced

   If \[a\] be the first term, \[r\] the common ratio, then sum \[{{S}_{n}}\] of first  \[n\] terms of a G.P. is given by   \[{{S}_{n}}=\frac{a(1-{{r}^{n}})}{1-r}\] and \[{{S}_{n}}=\frac{a-lr}{1-r}\],       (when \[|r|\,<1\])   \[{{S}_{n}}=\frac{a({{r}^{n}}-1)}{r-1}\] and \[{{S}_{n}}=\frac{lr-a}{r-1}\],       (when \[|r|\,>1\])   \[{{S}_{n}}=na\],   (when \[r=1\])

(1) When \[|r|\,<1\],    (or \[-1<r<1)\]; \[{{S}_{\infty }}=\frac{a}{1-r}\].   (2) If \[r\ge 1\], then \[{{S}_{\infty }}\] doesn't exist.

If \[a,G,b\] are in G.P., then G is called G.M. between \[a\] and \[b\].   (1) If \[a,\,{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}},\,b\] are in G.P. then \[{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}}\] are called n G.M.’s between \[a\] and \[b\].   (2) Insertion of geometric means : (i) Single G.M. between a and b : If a and b are two real numbers then single G.M. between \[a\] and \[b\]\[=\sqrt{ab}\].   (ii) n G.M.’s between a and b : If \[{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},\,......,\,{{G}_{n}}\] are n G.M.’s between a and b, then   \[{{G}_{1}}=ar=a{{\left( \frac{b}{a} \right)}^{\frac{1}{n+1}}}\], \[{{G}_{2}}=a{{r}^{2}}=a{{\left( \frac{b}{a} \right)}^{\frac{2}{n+1}}}\],   \[{{G}_{3}}=a{{r}^{3}}=a{{\left( \frac{b}{a} \right)}^{\frac{3}{n+1}}}\], ……………….., \[{{G}_{n}}=a{{r}^{n}}=a{{\left( \frac{b}{a} \right)}^{\frac{n}{n+1}}}\].

(1) If all the terms of a G.P. be multiplied or divided by the same non-zero constant, then it remains a G.P., with the same common ratio.   (2) The reciprocal of the terms of a given G.P. form a G.P. with common ratio as reciprocal of the common ratio of the original G.P.   (3) If each term of a G.P. with common ratio r be raised to the same power k, the resulting sequence also forms a G.P. with common ratio \[{{r}^{k}}\].   (4) In a finite G.P., the product of terms equidistant from the beginning and the end is always the same and is equal to the product of the first and last term. i.e., if \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......\,{{a}_{n}}\] be in G.P.   Then \[{{a}_{1}}\,{{a}_{n}}={{a}_{2}}\,{{a}_{n-1}}={{a}_{3}}\,{{a}_{n-2}}={{a}_{4}}\,{{a}_{n-3}}=..........={{a}_{r}}\,.\,{{a}_{n-r+1}}\]   (5) If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P. more...

A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.   Standard form : \[\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+....\]..   Example: The sequence \[1,\,\frac{1}{3},\,\frac{1}{5},\,\frac{1}{7},\,\frac{1}{9},...\] is a H.P., because the sequence 1, 3, 5, 7, 9, ….. is an A.P.

  If the H.P. be as \[\frac{1}{a},\,\frac{1}{a+d},\,\frac{1}{a+2d},\,....\] then corresponding A.P. is \[a,\,a+d,\,a+2d,\,.....\]   \[{{T}_{n}}\] of A.P. is \[a+(n-1)\,d\]   \[\therefore \] \[{{T}_{n}}\] of H.P. is \[\frac{1}{a+(n-1)\,d}\]   In order to solve the question on H.P., we should form the corresponding A.P. Thus, General term :   \[{{T}_{n}}=\frac{1}{a+(n-1)\,d}\] or \[{{T}_{n}}\text{ of H}\text{.P}\text{.}=\frac{1}{{{T}_{n}}\text{ of A}\text{.P}\text{.}}\].  

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.   Also, if a, H, b are in H.P., then H is called harmonic mean between \[a\] and \[b\].   (1) Insertion of harmonic means   (i) Single H.M. between \[a\] and \[b\]\[=\frac{2ab}{a+b}\].   (ii) H, H.M. of \[n\] non-zero numbers \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,....,\,{{a}_{n}}\]  is given by \[\frac{1}{H}=\frac{\frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+.....+\frac{1}{{{a}_{n}}}}{n}\].   (iii) Let \[a,\,\,b\] be two given numbers. If \[n\] numbers \[{{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}\] are inserted between \[a\] and \[b\] such that the sequence  \[a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b\] is a H.P., then \[{{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}\] are called \[n\] harmonic means between \[a\] and \[b\].   Now, \[a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b\] are in H.P.   \[\Rightarrow \] \[\frac{1}{a},\,\frac{1}{{{H}_{1}}},\,\frac{1}{{{H}_{2}}},\,......\frac{1}{{{H}_{n}}},\,\frac{1}{b}\] are in A.P.   Let \[D\] be the common difference of more...

  (1) No term of H.P. can be zero.            (2) If H is the H.M. between a and b, then   (i) \[\frac{1}{H-a}+\frac{1}{H-b}=\frac{1}{a}+\frac{1}{b}\]   (ii) \[(H-2a)(H-2b)={{H}^{2}}\]        (iii)          \[\frac{H+a}{H-a}+\frac{H+b}{H-b}=2\]  

  The combination of arithmetic and geometric progression is called arithmetico-geometric progression.  

  If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......,\,{{a}_{n}},\,......\] is an A.P. and \[{{b}_{1}},\,{{b}_{2}},\,\,......,\,{{b}_{n}},\,......\] is a G.P., then the sequence \[{{a}_{1}}{{b}_{1}},\,{{a}_{2}}{{b}_{2}},\,{{a}_{3}}{{b}_{3}},\]\[\,......,\,{{a}_{n}}{{b}_{n}},\,.....\] is said to be an arithmetico-geometric sequence.   Thus, the general form of an arithmetico geometric sequence is \[a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,(a+3d)\,{{r}^{3}},\,.....\]   From the symmetry we obtain that the nth term of this sequence is \[[a+(n-1)\,d]\,{{r}^{n-1}}\].   Also, let \[a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,(a+3d)\,{{r}^{3}},\,.....\]be an arithmetico-geometric sequence.   Then, \[a+\,(a+d)\,r\]\[+\,(a+2d)\,{{r}^{2}}+(a+3d)\,{{r}^{3}}+...\] is an arithmetico-geometric series.