Math:A.P.

The 1st,3rd and 9th terms of an A.P are the three terms of a G.P.if the 7th term of the A.P is 19.calculate the 20th term of the A.P and the sum of the first 12terms of the G.P(LEAVING YOUR ANSWER IN INDEX FORM)

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  1. for AP:
    term1 = a
    term3 = a+2d
    term9 = a+8d
    but term7 = 19 = a+6d
    so a = 19-6d

    so re-defining:
    term1 = 19-6d
    term3 = 19-6d+2d = 19-4d
    term9 = 19-6d + 8d = 19+2d

    these 3 are supposed to be a GP, so
    (19-4d)/(19-6d)= (19+2d)/(19-4d)
    361 - 152d + 16d^2 = 361 -76d - 12d^2
    28d^2 - 76d = 0
    d(28d - 76) = 0
    d = 0 or d = 76/28 = 19/7

    case1 (trivial case) , d = 0
    then all terms in the AP would be 19
    i.e. 19 19 19 19 ...
    of course the first 3 would be a GP also , etc

    case 2:
    d = 19/7
    a = 19 - 6d = 19-6(19/7) = 19/7

    for AP, term 20 = a+19d = 19/7+19(19/7) = 380/7

    now term1 of AP = term1 of GP
    a of GP = 19/7
    term 3 of AP = term2 of GP
    term2 of GP = a+2d = 19/7 + 2(19/7) = 57/7
    so r of GP = (57/7) รท (19/7) = 3

    sum 12 of GP = a(r^12 - 1)/r
    = (19/7)(3^12 - 1)/2

    check my arithmetic

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  2. the 4th & 9th term of the A.P are 1 & 19 respectively. calculate the 1st term

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  3. Solve this problem

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  4. Please tell this with clear explanations

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  5. Ssss

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  6. im so

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  7. Right ๐Ÿ‘๐Ÿ‘

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