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In each of the following APs the 1st, 2nd and last terms are given. Find the common difference the number of terms in each sequence.

Mathematics

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Here are the common differences and number of terms for each arithmetic progression:

The general formula for the nn-th term of an arithmetic progression (AP) is an=a1+(n−1)da_n = a_1 + (n-1)d, where a1a_1 is the first term, dd is the common difference, and nn is the number of terms. The common difference dd can be found using d=a2−a1d = a_2 - a_1.

Part a· 2 steps

1, 5, ..., 69

  1. Find the common difference dd.

    d=a2−a1=5−1=4d = a_2 - a_1 = 5 - 1 = 4
  2. Find the number of terms nn.

    an=a1+(n−1)da_n = a_1 + (n-1)d 69=1+(n−1)469 = 1 + (n-1)4 68=(n−1)468 = (n-1)4 684=n−1\frac{68}{4} = n-1 17=n−117 = n-1 n=17+1n = 17 + 1 n=18n = 18

    The common difference is 4\boxed{4} and the number of terms is 18\boxed{18}.

Part b· 2 steps

80, 83, ..., 152

  1. Find the common difference dd.

    d=a2−a1=83−80=3d = a_2 - a_1 = 83 - 80 = 3
  2. Find the number of terms nn.

    an=a1+(n−1)da_n = a_1 + (n-1)d 152=80+(n−1)3152 = 80 + (n-1)3 152−80=(n−1)3152 - 80 = (n-1)3 72=(n−1)372 = (n-1)3 723=n−1\frac{72}{3} = n-1 24=n−124 = n-1 n=24+1n = 24 + 1 n=25n = 25

    The common difference is 3\boxed{3} and the number of terms is 25\boxed{25}.

Part c· 2 steps

98, 91, ..., 0

  1. Find the common difference dd.

    d=a2−a1=91−98=−7d = a_2 - a_1 = 91 - 98 = -7
  2. Find the number of terms nn.

    an=a1+(n−1)da_n = a_1 + (n-1)d 0=98+(n−1)(−7)0 = 98 + (n-1)(-7) −98=(n−1)(−7)-98 = (n-1)(-7) −98−7=n−1\frac{-98}{-7} = n-1 14=n−114 = n-1 n=14+1n = 14 + 1 n=15n = 15

    The common difference is −7\boxed{-7} and the number of terms is 15\boxed{15}.

Part d· 2 steps

2.3, 3.4, ..., 21

  1. Find the common difference dd.

    d=a2−a1=3.4−2.3=1.1d = a_2 - a_1 = 3.4 - 2.3 = 1.1
  2. Find the number of terms nn.

    an=a1+(n−1)da_n = a_1 + (n-1)d 21=2.3+(n−1)1.121 = 2.3 + (n-1)1.1 21−2.3=(n−1)1.121 - 2.3 = (n-1)1.1 18.7=(n−1)1.118.7 = (n-1)1.1 18.71.1=n−1\frac{18.7}{1.1} = n-1 17=n−117 = n-1 n=17+1n = 17 + 1 n=18n = 18

    The common difference is 1.1\boxed{1.1} and the number of terms is 18\boxed{18}.

Part e· 2 steps

6, -212\frac{1}{2}, ..., 71

  1. Find the common difference dd.

    Convert −212-2\frac{1}{2} to a decimal: −2.5-2.5.

    d=a2−a1=−2.5−(−6)=−2.5+6=3.5d = a_2 - a_1 = -2.5 - (-6) = -2.5 + 6 = 3.5
  2. Find the number of terms nn.

    an=a1+(n−1)da_n = a_1 + (n-1)d 71=−6+(n−1)3.571 = -6 + (n-1)3.5 71+6=(n−1)3.571 + 6 = (n-1)3.5 77=(n−1)3.577 = (n-1)3.5 773.5=n−1\frac{77}{3.5} = n-1 22=n−122 = n-1 n=22+1n = 22 + 1 n=23n = 23

    The common difference is 3.5\boxed{3.5} and the number of terms is 23\boxed{23}.

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