If $$a1$$,$$a2$$,…an are in H.P. then the expression $$a1a2+a2a3+$$…$$+an$$−$$1an$$ is equal to

Question

If $$a1$$,$$a2$$,…an are in H.P. then the expression $$a1a2+a2a3+$$…$$+an$$−$$1an$$ is equal to

Infinity Learn · Accepted Answer

$$a1$$,$$a2$$,…an are in H.P.⇒ $$1a1$$,$$1a2$$,…$$1an$$ are in A.P.⇒ $$1a2$$−$$1a1=1a3$$−$$1a2=$$⋯$$1an$$−$$1an$$−$$1=d(say)$$⇒ $$a1a2=1da1$$−$$a2$$,$$a2a3=1da2$$−$$a3$$,…an−$$1an=1dan$$−$$1$$−anThus,   $$a1a2+a2a3+$$…$$+an$$−$$1an$$                         $$=1da1$$−$$a2+a2$$−$$a3+$$…$$+an$$−$$1$$−an              $$=1da1$$−anBut    $$1an=1a1+(n$$−$$1)d$$⇒$$a1$$−$$anana1=(n$$−$$1)d$$∴ $$a1a2+a2a3+$$…$$+an$$−$$1an=(n$$−$$1)a1an$$

If $a_{1}, a_{2}, \dots a_{n}$ are in H.P. then the expression $a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}$ is equal to

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If a1,a2,…an are in H.P. then the expression a1a2+a2a3+…+an−1an is equal to

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If $a_{1}, a_{2}, \dots a_{n}$ are in H.P. then the expression $a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}$ is equal to