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The first and last terms of an Arithmetic progression (AP) are -3 and 393 respectively. If the progression has 100 terms, find the 33rd term of the AP.

Mathematics

ScanSolve AI Answer

Answer

125

Step-by-step solution

3 steps
  1. Identify the given values and the formula for the nn-th term of an Arithmetic Progression (AP).

    The first term, a=−3a = -3.
    The last term, L=393L = 393.
    The total number of terms, n=100n = 100.
    The formula for the nn-th term is Tn=a+(n−1)dT_n = a + (n-1)d, where dd is the common difference.

  2. Use the last term to find the common difference (dd).

    The last term is the 100th term, so T100=393T_{100} = 393.
    Substitute the values into the formula:

    393=−3+(100−1)d393 = -3 + (100-1)d 393=−3+99d393 = -3 + 99d

    Add 3 to both sides:

    393+3=99d393 + 3 = 99d 396=99d396 = 99d

    Divide by 99 to find dd:

    d=39699d = \frac{396}{99} d=4d = 4
  3. Calculate the 33rd term (T33T_{33}).

    Now that we have the common difference d=4d=4, we can find the 33rd term using the formula Tn=a+(n−1)dT_n = a + (n-1)d with n=33n=33.

    T33=−3+(33−1)(4)T_{33} = -3 + (33-1)(4) T33=−3+(32)(4)T_{33} = -3 + (32)(4) T33=−3+128T_{33} = -3 + 128 T33=125T_{33} = 125

    The 33rd term of the AP is 125\boxed{125}.

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