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\text { If } a_{1}, a_{2}, a_{3}, \dots \dots , a_{n} are in H.P  then a_{1} a_{2}+a_{2} a_{3}+\ldots \dots \dots+a_{n-1} a_{n} will be equal to 

Option: 1

a_{1} a_{n}


Option: 2

na_{1} a_{n}


Option: 3

(n-1)a_{1} a_{n}


Option: 4

None of these


Answers (1)

best_answer

General term of a Harmonic Progression

The nth term or general term of a H.P. is the reciprocal of the nth term of the corresponding A.P. Thus, if a_1,a_2,a_3,......,a_n is an H.P. and the common difference of corresponding A.P. is d, i.e. d=\frac{1}{a_n}-\frac{1}{a_{n-1}}, then the general term or nth term of an H.P. is given by \mathrm{a_n=\frac{1}{\frac{1}{a_1}+(n-1)d}}

 

Now,

a_{1}, a_{2}, a_{3}, \dots \dots , a_{n} \text{are in H.P.}\\ \frac{1}{a_{2}}-\frac{1}{a_1}=\frac{1}{a_{3}}-\frac{1}{a_2}.......=\frac{1}{a_{n}}-\frac{1}{a_{n-1}}=d\\ \frac{a_{1}-a_{2}}{a_1 \cdot a_{2}}=\frac{a_{2}-a_{3}}{a_2 \cdot a_{3}}.....=\frac{a_{n-1}-a_{n}}{a_n \cdot a_{n-1}}=d\\ {a_{1}-a_{2}}=d\ a_1 \cdot a_{2},\ \ {a_{2}-a_{3}}=d\ a_2 \cdot a_{3},\ \ ....\ \ {a_{n-1}-a_{n}}=d\ a_n \cdot a_{n-1}\\ Adding\,\,all\,\,these\,\,\\{a_{1}-a_{2}}+{a_{2}-a_{3}}+...+ {a_{n-1}-a_{n}}=d(a_1 \cdot a_{2}+a_2 \cdot a_{3}\ +... +a_{n-1} \cdot a_{n} )\\ a_1 - a_{n}=d(a_1 \cdot a_{2}+a_2 \cdot a_{3}\ +... +a_{n-1} \cdot a_{n} ) . . . (i)\\ \text{nth term of H.P.}\\ \frac{1}{a_n}=\frac{1}{a_1}+(n-1)d\\ \frac{1}{a_n}-\frac{1}{a_1}=(n-1)d\\ \frac{ a_{1}-a_{n}}{a_{1}a_{n}} =(n-1)d . . . (ii)\\ \text{by equation (i) and (ii)}\\ (a_1 \cdot a_{2}+a_2 \cdot a_{3}\ +... +a_{n-1} \cdot a_{n} )=(n-1)a_{1} a_{n}

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SANGALDEEP SINGH

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