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  • If the nth terms of the two APs: 9, 7, 5, ... and 24, 21, 18,... are the same, find the value of n. Also find that term.

If the nth terms of the two APs: 9, 7, 5, ... and 24, 21, 18,... are the same, find the value of n. Also find that term.

Answers (1)

The given APs are

 9, 7, 5, \ldots \quad \text{and} \quad 24, 21, 18, \ldots

  In AP 

9, 7, 5, $ \ldots $ ..\\

a_{1} = a = 9, \quad a_{2} = 7 \\ d_{1} = a_{2} - a_{1} = 7 - 9 = -2 \\ a_{n} = a + (n - 1)d_{1} \quad (\text{1}) \\ \text{In the AP } 24, 21, 18, \ldots \\ b_{1} = b = 24, \quad b_{2} = 21 \\ d_{2} = b_{2} - b_{1} = 21 - 24 = -3 \\ b_{n} = b + (n - 1)d_{2} \quad (\text{2})

Here the number of terms n is equal in both case 

  According to question

 a\textsubscript{n} = b\textsubscript{n}\\

9 + (n - 1)(-2) = 24 + (n - 1)(-3) \\ 9 - 2n + 2 = 24 - 3n + 3 \\ 9 - 2n + 2 - 24 + 3n - 3 = 0 \\ n - 16 = 0 \\ n = 16 \\ a_{16} = a + (16 - 1)d \quad (\text{using } a_{n} = a + (n-1)d) \\ = 9 + 15(-2) \\ = 9 - 30 = -21 \\ a_{16} = -21 \\ \text{The } n\textsuperscript{th} \text{ term is } -21 \text{ and the value of } n \text{ is } 16

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