the sum of first six terms of an AP is 42. The ratio of 10th term to its 38th term is 1:3.calculate the fist term and 13th term of the AP .​

Question

the sum of first six terms of an AP is 42. The ratio of 10th term to its 38th term is 1:3.calculate the fist term and 13th term of the AP .​

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Alexandra 1 month 2021-08-18T11:45:08+00:00 2 Answers 0 views 0

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    0
    2021-08-18T11:46:43+00:00

     Let \: a \: and \: d \: are \: first \:term \:and

     common \: difference \: of \: an \: A.P

     Sum \: of \: n \:terms (S_{n} ) = \frac{n}{2} [ 2a + (n-1)d ]

     Here , n = 6 \:and\: S_{6} = 42

     \implies \frac{6}{2}[ 2a +5d ] = 42

     \implies 3(2a+5d) = 42

     \implies 2a + 5d = \frac{42}{3}

     \implies 2a + 5d = 14 \: --(1)

    /* We know that, */

     \boxed{ \blue{n^{th} \:term (a_{n}) = a + (n-1)d }}

     Ratio \: of \: 10^{th} \: term \: and

     38^{th} \:term\: = \frac{1}{3}

     \implies \frac{a+9d}{a+37d} = \frac{1}{3}

     \implies 3(a+9d) = a+37d

     \implies 3a+ 27d= a+37d

     \implies 3a - a = 37d - 27d

     \implies 2a = 10d \: --(2)

     Put \: 2a = 10d \: in \: equation \: (1), we \:get

     \implies 10d + 5d = 14

     \implies 15d = 14

     \implies d = \frac{14}{15} \: --(3)

     Put \: d = \frac{14}{15}\: in\: equation \:(2) ,

     we \:get

     2a = 10 \times \frac{14}{15}

     \implies a = 2 \times \frac{7}{3}

     \implies a =  \frac{14}{3} \: --(4)

     13^{th} \:term (a_{13} ) = a + 12d

     =  \frac{14}{3} + 12 \times \frac{14}{15}

     = \frac{14}{3} + 4 \times \frac{14}{5}

     = \frac{14}{3} +  \frac{56}{5}

     = \frac{70+168}{15}

     = \frac{238}{15}

    •••♪

    0
    2021-08-18T11:46:53+00:00

    Answer: First-term = 14/3, 13th term = 238/15

    Step-by-step explanation:

    The sum of the first 6 terms = 42

    Sum = (n/2)(2a+(n-1)d)\\Here, n = 6, sum = 42\\42 = (6/2)(2a+5d)\\2a+5d = 14...........(1)

    Ratio of 10th and 38th term = 1:3

    (a+9d)/(a+37d) = 1/3\\a = 5d.........(2)

    From (1) and (2);

    a = 14/3 = First term

    d = 14/15

    So, 13th term = a+12d = 238/15

    (I hope it will be marked as Brainliest)

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