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State True/False.

If 4 times the 4th term of an AP is equal to 18 times its 18 th term then its 22nd term is 0.

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True

False

answer is A.

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Detailed Solution

Given that, 4 times the 4th term of an AP is equal to 18 times its 18 th term.
According to the given condition:

4 × t 4 = 18 × t 18 − − − − (1)

Let a be the first term and d be the common difference of the A.P.
Formula for the

n t h

term:

t n = a + n − 1 × d

Applying the formula of

n t h

term of AP and calculate the 4th term.

∴ t 4 = a + (4 − 1) d ⇒ t 4 = a + 3 d − − − − − (2)

Applying the formula of

n t h

term of AP and calculate the 18th term.

∴ t 18 = a + (18 − 1) d ⇒ t 18 = a + 17 d − − − − − (3)

Applying the formula of

n t h

term of AP and calculate the 22th term.

∴ t 22 = a + (22 − 1) d − − − − − (4)

Put the value of

t 4

from equation (2) and the value of

t 18

from equation (3) into equation (1):

∴ 4 (a + 3 d) = 18 (a + 17 d) ⇒ 4 a + 12 d = 18 a + 306 d ⇒ − 14 a = 294 d ⇒ a = − 21 d

⇒ a + 21 d = 0 ⇒ a + (22 − 1) d = 0 − − − − − (5)

Put

a + (22 − 1) d = t 22 from equation (4) in equation (5)

, Thus,

t 22 = 0

.
Therefore, 22nd term of A.P. is 0.
Thus, the given statement is true.
Hence, option (1) is correct.

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State True/False.If 4 times the 4th term of an AP is equal to 18 times its 18 th term then its 22nd term is 0.

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State True/False.

If 4 times the 4th term of an AP is equal to 18 times its 18 th term then its 22nd term is 0.