Question

The ninth term of an A.P. is equal to the seventh times the second term and the twelfth term exceeds five times the third term by 2. Find the first term and the common difference?

Answer 1

Answer:

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

Given,

a₉ = 7a₂

or, a+8d = 7(a+d)  ________(i)

and

a₁₂  = 5a₃ + 2

or, a+11d = 5(a+2d) + 2   ________(ii)

from eq. (i)

a+8d = 7a+7d

-6a + d = 0   ________(iii)

from eq. (ii)

a+11d = 5a+10d + 2

-4a + d = 0   ________(iv)

Substracting eq. (iv) from eq. (iii), we get,

-2a = 2

a = 1

or,

from eq. (iii)

-6a + d = 0

d = 6

Hence,

First term is 1

Common difference is 6

Answer 2

Answer :

• First term, a = 1
• Common difference, d = 6

Solution :

Given :

• The ninth term of an A.P. is equal to the seven times the second term.
• The twelfth term exceeds five times the third term by 2.

To Find :

• First term (a).
• Common difference (d).

Let's solve,

It is given that ,

• 9th term is 7 times the second term.

So,

$\:\:\:\:\: \red\bigstar\sf T_9 = 7\times T_2$

It can be also written as :

$\:\:\:\:\: \bf a+8d = 7(a+d) ……(i)$

Solving (i) ,

$\implies \sf a + 8d = 7(a+d)\\ \sf \implies a + 8d = 7a + 7d \\\sf \implies 8d - 7d = 7a - a \\\implies\sf \pink{d=6a} ……(ii)$

━━━━━━━━━━━━

• 12th term exceeds 5 times the 3rd term by 2.

So,

$\:\:\:\:\: \red\bigstar\sf T_{12} = 5\times T_3 + 2$

It can be also written as :

$\:\:\:\:\: \bf a+11d = 5(a+2d) + 2……(iii)$

Solving (iii) ,

$\implies\sf a+11d = 5(a+2d) + 2 \\\implies\sf a + 11d = 5a+10d+2 \\\implies\sf 11d-10d = 5a-a+2 \\\implies\sf \pink{d = 4a+2 }……(iv)$

$\sf \large\color{blue} \mathfrak{ Substituting\: (ii)\: in\: (iv) \:}$

• d=6a ……(ii)

• d = 4a+2 ……(iv)

$\implies \sf d = 4a+2 \\\implies\sf 6a = 4a+2 \\\implies\sf 2a= 2 \\\implies\sf \underline{\boxed{\mathfrak{\pink{a=1}}}}$

$\sf \large\color{blue} \mathfrak{ Putting\:the \:vaule\:of\: a\: in\: (iii) \:}$

• d = 6a

$\implies\sf d=6(1) \\\implies\underline{\boxed{\mathfrak{\pink{d=6}}}}$

Hence,

First term(a) is 1 and Common difference(d) is 6.

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