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Prove that there are 17 identical terms in the two A.P.'s 2, 5

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Prove that there are $17$ identical terms in the two A.P.'s $2, 5, 8, 11,$ _ $60$ terms and $3, 5, 7, 9$ , __ $50$ terms.

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Prove that there are

17

identical terms in the two A.P.'s

2, 5, 8, 11,

_______

60

terms and

3, 5, 7, 9

, ________

50

terms.

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Text solutionVerified

Let

T_{p} = T_{q} \Rightarrow 3 p - 1 = 2 q + 1

Subtract

5

from both sides

∴ 3 (p - 2) = 2 (q - 2)

\frac{p - 2}{2} = \frac{q - 2}{3} = k

, say

∴ p = 2 k + 2

and

q = 3 k + 2

Now p varies from

1

60

and q varies from

1

50

Hence we have the following

1 \leq 2 k + 2 \leq 60

and

1 \leq 3 k + 2 \leq 50

∴ - \frac{1}{2} \leq k \leq 29

and

- \frac{1}{3} \leq k \leq 16

Clearly

k = 0, 1, 2, 3, ..., 16

for common values and hence there will be

17

common terms.

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Question Text	Prove that there are $17$ identical terms in the two A.P.'s $2, 5, 8, 11,$ _______ $60$ terms and $3, 5, 7, 9$ , ________ $50$ terms.
Answer Type	Text solution:1
Upvotes	150

Prove that there are 17 identical terms in the two A.P.'s 2,5,8,11,_______60 terms and 3,5,7,9, ________50 terms.

Text solutionVerified

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Prove that there are $17$ identical terms in the two A.P.'s $2, 5, 8, 11,$ _ $60$ terms and $3, 5, 7, 9$ , __ $50$ terms.