if the sum of the 3rd and 7th term of an arithmetic progression is 6 and their product is 8. Find the sum of first 20 terms of an arithmetic

Question

if the sum of the 3rd and 7th term of an arithmetic progression is 6 and their product is 8. Find the sum of first 20 terms of an arithmetic progression​

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Allison 1 month 2021-08-23T00:52:04+00:00 1 Answer 0 views 0

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    2021-08-23T00:53:51+00:00

    ANSWER

    Let a and d be the first term and common difference of AP

    nth term of AP

    a

    n

    =a+(n−1)d

    ∴a

    3

    =a+(3−1)d=a+2d

    a

    7

    =a+(7−1)d=a+6d

    Given a

    3

    +a

    7

    =6

    ∴(a+2d)+(a+6d)=6

    ⇒2a+8d=6

    ⇒a+4d=3….(1)

    Also given

    a

    3

    ×a

    7

    =8

    ∴(a+2d)(a+6d)=8

    ⇒(3−4d+2d)(3−4d+6d)=8 [Using (1)]

    ⇒(3−2d)(3+2d)=8

    ⇒9−4d

    2

    =8

    ⇒4d

    2

    =1

    ⇒d

    2

    =

    4

    1

    ⇒d=±

    2

    1

    When d=

    2

    1

    a=3−4d=3−4×

    2

    1

    =3−2=1

    When d=−

    2

    1

    a=3−4d=3+4×

    2

    1

    =3+2=5

    When a=1 & d=

    2

    1

    S

    16

    =

    2

    16

    [2×1+(16−1)×

    2

    1

    ]=8(2+

    2

    15

    )=4×19=76

    When a=5 & d=−

    2

    1

    S

    16

    =

    2

    16

    [2×5+(16−1)×(−

    2

    1

    )]=8(10−

    2

    15

    )=4×5=20

    Thus, the sum of first 16 terms of the AP is 76 or 20.

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