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1. If the 5th term of an arithmetic progression (AP) is 10 and the 10th term is 5, find:

a) The first term.

b) The common difference.

2. If the 12th term of an AP is double the 5th term, find the common difference given that the first term is 7.

Answer :

To solve these problems related to Arithmetic Progressions (AP), we will use the definition of the nth term of an AP.

An Arithmetic Progression is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the common difference, denoted as [tex]d[/tex]. The general formula for the nth term [tex]a_n[/tex] of an AP is:

[tex]a_n = a + (n-1)\cdot d[/tex]

where [tex]a[/tex] is the first term and [tex]d[/tex] is the common difference.

Let's break down the given problems:

  1. If the 5th term of an AP = 10 and the 10th term = 5, find:

    a) The first term

    Given: [tex]a_5 = 10[/tex] and [tex]a_{10} = 5[/tex].

    For the 5th term: [tex]a + 4d = 10[/tex] (Equation 1)

    For the 10th term: [tex]a + 9d = 5[/tex] (Equation 2)

    We have two equations:

    [tex]a + 4d = 10[/tex]
    [tex]a + 9d = 5[/tex]

    By subtracting Equation 1 from Equation 2:

    [tex](a + 9d) - (a + 4d) = 5 - 10[/tex]

    [tex]5d = -5[/tex]

    Solving for [tex]d[/tex]:

    [tex]d = -1[/tex]

    Substituting [tex]d = -1[/tex] into Equation 1:

    [tex]a + 4(-1) = 10[/tex]

    [tex]a - 4 = 10[/tex]

    [tex]a = 14[/tex]

    So, the first term [tex]a = 14[/tex] and the common difference [tex]d = -1[/tex].

  2. If the 12th term of an AP is double the 5th term, find the common difference given that the first term is 7.

    Given: [tex]a_1 = 7[/tex], [tex]a_{12} = 2a_5[/tex].

    The formula for the 12th term is:
    [tex]a + 11d = a_{12}[/tex]

    The formula for the 5th term is:
    [tex]a + 4d = a_5[/tex]

    Since [tex]a_{12} = 2a_5[/tex]:
    [tex]a + 11d = 2(a + 4d)[/tex]

    Substituting [tex]a = 7[/tex]:
    [tex]7 + 11d = 2(7 + 4d)[/tex]

    Simplifying:
    [tex]7 + 11d = 14 + 8d[/tex]

    [tex]11d - 8d = 14 - 7[/tex]

    [tex]3d = 7[/tex]

    Solving for [tex]d[/tex]:
    [tex]d = \frac{7}{3}[/tex]

Thus, the common difference [tex]d[/tex] is [tex]\frac{7}{3}[/tex].

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