# 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. - Mathematics

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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.

#### Solution

Let the first term, common difference

And number of terms of the AP: 9, 7, 5,…. are

a1, d1 and n1, respectively

i.e., first term (a1) = 9

And common difference (d1) = 7 – 9 = –2

⇒ T"'"_(n_1) = a_1 + (n_1 - 1)d_1

⇒ T"'"_(n_1) =9 + (n_1 - 1)(-2)

⇒ T"'"_(n_1) = 9 - 2n_1 + 2

⇒ T"'"_(n_1) = 11 - 2n_1   ......(i)

∵ nth term of an AP, Tn = a + (n – 1)d

Let the first term, common difference and the number of terms of the AP: 24, 21, 18, … are a2, d2 and n2, respectively

i.e., first term, (a2) = 24

And common difference (d2) = 21 – 24 = – 3

∴ Its nth term, T"'"_(n_2) = a_2 + (n_2 - 1)d_2

⇒ T"'"_(n_2) = 24 + (n_2 - 1)(-3)

⇒ T"'"_(n_2) = 24 - 3n_2 + 3

⇒ T"'"_(n_2) = 27 - 3n_2  .....(ii)

Now, by given condition,

nth terms of the both APs are same

i.e., 11 - 2n_1 = 27 - 3n_2   ......[From equations (i) and (ii)]

⇒ n = 16  ......[because n_1 - n_2 = n]

∴ nth terms of first AP,

T"'"n_1 = 11 - 2n_1

= 11 - 2(16)

= 11 - 32

= – 21

And nth terms of second AP,

T"'"n_2 = 27 - 3n_2

= 27 - 3(16)

= 24 - 48

= – 21

Hence, the value of n is 16 and that term i.e., nth term is – 21.

Concept: Arithmetic Progression
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10
Chapter 5 Arithematic Progressions
Exercise 5.3 | Q 14 | Page 53
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