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Question Number 32745 by vakil vishaWakrma last updated on 01/Apr/18

The first, second and middle terms of an AP are a, b, c respectively. Their sum is

$$\mathrm{The}\:\mathrm{first},\:\mathrm{second}\:\mathrm{and}\:\mathrm{middle}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{AP}\:\mathrm{are}\:{a},\:{b},\:{c}\:\mathrm{respectively}.\:\mathrm{Their}\:\mathrm{sum}\:\mathrm{is} \\ $$

Answered by Rasheed.Sindhi last updated on 01/Apr/18

Let the AP has n (odd) terms and common difference d. d=b−a The middle term=(((n+1)/2))th term=c c=a+(((n+1)/2)−1)(b−a) ((n−1)/2)=((c−a)/(b−a)) n=((2(c−a))/(b−a))+1 =((2c−2a+b−a)/(b−a)) =((−3a+b+2c)/(b−a)) Sum=(middle-term)×(number of terms) =c×n=c(((−3a+b+2c)/(b−a))) =(c/(b−a))(−3a+b+2c)

$$\mathrm{Let}\:\mathrm{the}\:\mathrm{AP}\:\mathrm{has}\:\mathrm{n}\:\left(\mathrm{odd}\right)\:\mathrm{terms}\:\mathrm{and}\:\mathrm{common} \\ $$$$\mathrm{difference}\:\mathrm{d}. \\ $$$$\mathrm{d}=\mathrm{b}−\mathrm{a} \\ $$$$\mathrm{The}\:\mathrm{middle}\:\mathrm{term}=\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{2}}\right)\mathrm{th}\:\mathrm{term}=\mathrm{c} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{c}=\mathrm{a}+\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right)\left(\mathrm{b}−\mathrm{a}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{n}−\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{c}−\mathrm{a}}{\mathrm{b}−\mathrm{a}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}=\frac{\mathrm{2}\left(\mathrm{c}−\mathrm{a}\right)}{\mathrm{b}−\mathrm{a}}+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{2c}−\mathrm{2a}+\mathrm{b}−\mathrm{a}}{\mathrm{b}−\mathrm{a}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{−\mathrm{3a}+\mathrm{b}+\mathrm{2c}}{\mathrm{b}−\mathrm{a}} \\ $$$$\mathrm{Sum}=\left(\mathrm{middle}-\mathrm{term}\right)×\left(\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\mathrm{c}×\mathrm{n}=\mathrm{c}\left(\frac{−\mathrm{3a}+\mathrm{b}+\mathrm{2c}}{\mathrm{b}−\mathrm{a}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{c}}{\mathrm{b}−\mathrm{a}}\left(−\mathrm{3a}+\mathrm{b}+\mathrm{2c}\right) \\ $$

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