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Grade 12Algebra

If an is a se sequence in HP with a2 is equal to 1/2 a5 is equal to 1/11 then find the value of
1/a1+1/a2+1/a3.........1/a20

Question image for If a n is a se sequence in HP with a 2 is equal t
Profile image of Samidha
5 Years agoGrade 12
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1 Answer

Profile image of Lord 0p
5 Years ago

 

,

 

Given:
a1​,a2​,a3​,....,an​   are in H.P.
∴a2​=a1​+a3​2a1​a3​​                                                    .....( 1 )
⇒a1​a2​+a2​a3​=2a1​a3​
⇒a1​a2​+a2​a3​=(3−1)a1​a3​                           .....( 2 )
Now,
∴a3​=a2​+a4​2a2​a4​​
⇒a3​(a2​+a4​)=2a4​.a1​+a3​2a1​a3​​                         [From eq ( 1 )]
⇒(a1​+a3​)(a2​+a4​)=4a1​a4​
⇒a1​a2​+a2​a3​+a3​a4​=(4−1)a1​a4​                .....( 3 )
 
Comparing eq (2) and (3) for upto n terms:
⇒a1​a2​+a2​a3​+a3​a4​+....+an−1​an​=(n−1)a1​an​
 
,
 

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