which term of ap -7,-12,-17,-22,... will be -88? is -100 any term of the ap? give reason for your answer
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which term of ap -7,-12,-17,-22,... will be -88? is -100 any term of the ap? give reason for your answer
Give Detailed Answer
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Step-by-step explanation:
Given AP series is - 7, - 12, - 17, - 22, ...
So, we have
First term, a = - 7
Common difference, d = - 12 - (- 7) = -12 + 7 = - 5
Now, let assume that nth term of an AP be - 88
So, we have
[tex]\sf \: a_n = - 88 \\ \\ [/tex]
[tex]\sf \: a + (n - 1)d = - 88 \\ \\ [/tex]
[tex]\sf \: - 7 - 5 (n - 1) = - 88 \\ \\ [/tex]
[tex]\sf \: - 7 - 5 n + 5 = - 88 \\ \\ [/tex]
[tex]\sf \: - 2 - 5 n = - 88 \\ \\ [/tex]
[tex]\sf \: - 5 n = - 88 + 2 \\ \\ [/tex]
[tex]\sf \: - 5 n = - 86 \\ \\ [/tex]
[tex]\sf \: n = \dfrac{86}{5} \: \: which \: is \: not \:an \: integer \\ \\ [/tex]
[tex]\sf\implies \: - 88 \: is \: not \: term \: of \: the \: series \: - 7, \: - 12, \: - 17, \: ... \\ \\ \\ [/tex]
Now, let further assume that mth term of an AP series is - 100.
So, we have
[tex]\sf \: a_m = - 100 \\ \\ [/tex]
[tex]\sf \: a + (m - 1)d = - 100 \\ \\ [/tex]
[tex]\sf \: - 7 - 5 (m - 1) = - 100 \\ \\ [/tex]
[tex]\sf \: - 7 - 5m + 5 = - 100 \\ \\ [/tex]
[tex]\sf \: - 2 - 5m = - 100 \\ \\ [/tex]
[tex]\sf \: - 5m = - 100 + 2\\ \\ [/tex]
[tex]\sf \: - 5m = - 98\\ \\ [/tex]
[tex]\sf \: m = \dfrac{98}{5} \: \: which \: is \: not \:an \: integer \\ \\ [/tex]
[tex]\sf\implies \: - 100 \: is \: not \: term \: of \: the \: series \: - 7, \: - 12, \: - 17, \: ... \\ \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formula Used
↝ nᵗʰ term of an arithmetic progression is,
[tex]\begin{gathered}\bigstar\:\:{\underline{{\boxed{\bf{{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}[/tex]
Wʜᴇʀᴇ,
a is the first term of the progression.
n is the no. of terms.
d is the common difference.