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the fifth term of an arithmetic sequence is 24 and the ninth term is 52. Write an equation that can be used to find the nth term of this sequence



Answer :

Answer:

The sequence is [tex]\bf U_n=7n-11[/tex].

Step-by-step explanation:

To find the nth term of this sequence, we can use the arithmetic sequence formula:

[tex]\boxed{U_n=U_1+(n-1)d}[/tex]

where:

  • [tex]U_n[/tex] = nth term
  • [tex]U_1[/tex] = 1st term
  • [tex]n[/tex] = total number of terms
  • [tex]d[/tex] = difference for each pair of consecutive terms

Given:

  • [tex]U_5[/tex] = 24
  • [tex]U_9[/tex] = 52

[tex]U_5=U_1+(5-1)d[/tex]

[tex]24=U_1+4d\ ...\ [1][/tex]

[tex]U_9=U_1+(9-1)d[/tex]

[tex]52=U_1+8d\ ...\ [2][/tex]

Combining [1] & [2]:

[tex]24=U_1+4d\ \Longleftrightarrow\ 2U_1+8d=48[/tex]

[tex]52=U_1+8d\ \Longleftrightarrow\ U_1+8d=52[/tex]

------------------------------------------------- (-)

                              [tex]\bf U_1=-4[/tex]

Substitute the value of [tex]U_1[/tex] into [1]:

[tex]24=U_1+4d[/tex]

[tex]24=-4+4d[/tex]

[tex]4d=24+4[/tex]

[tex]d=28\div 4[/tex]

[tex]\bf d=7[/tex]

Substitute the values of [tex]U_1[/tex] and [tex]d[/tex] into the formula, then we have the equation of this sequence:

[tex]U_n=U_1+(n-1)d[/tex]

[tex]U_n=-4+(n-1)(7)[/tex]

[tex]U_n=-4+7n-7[/tex]

[tex]\bf U_n=7n-11[/tex]

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