Question

The nth term of an ap is equal to 7 times the 2nd term and twenty first and exceeds5times the third term By 2 fine the 1st term and a common difference
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Answer 1

Given :-

The ninth term of an AP is equal to seven times the second term.

Twelfth term exceeds five times the third term by 2.

To Find :-

a = ?? and d = ??

Solution :-

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

Given, a(9) = 7a(2)

or, a + 8d = 7(a + d)  .... (i)

And, a(12) = 5a(3) + 2

Again, a + 8d = 5(a + 2d) + 2 ...(ii)

From (i), a + 8d = 7a + 7d

⇒ - 6a + d = 0 .... (iii)

From (ii), a + 11d = 5a + 10d + 2

⇒ - 4a + d = 2 .... (iv)

Subtracting (iv) from (iii), we get

⇒ - 2a = - 2

⇒ a = 2/2

⇒ a = 1

From (iii),

⇒ - 6 + d = 0

⇒ d = 6

Hence, the first term is 1 and the common difference is 6.

Answer 2

Given :-

The ninth term of an AP is equal to seven times the second term.

Twelfth term exceeds five times the third term by 2.

To Find :-

a = ?? and d = ??

Solution :-

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

Given, a(9) = 7a(2)

or, a + 8d = 7(a + d)  .... (i)

And, a(12) = 5a(3) + 2

Again, a + 8d = 5(a + 2d) + 2 ...(ii)

From (i), a + 8d = 7a + 7d

⇒ - 6a + d = 0 .... (iii)

From (ii), a + 11d = 5a + 10d + 2

⇒ - 4a + d = 2 .... (iv)

Subtracting (iv) from (iii), we get

⇒ - 2a = - 2

⇒ a = 2/2

⇒ a = 1

From (iii),

⇒ - 6 + d = 0

⇒ d = 6

Hence, the first term is 1 and the common difference is 6.