Question

the 8th term of an AP is 32 and its 12tj term is 53 the first term and common difference is​

Answer 1

Hint: Recall the concept of general form of a term of an AP, you will get two equations and you can solve the two equations formed by using elimination or substitution method for finding first term and common difference.

Solution :-

By the given information, we have :

• 8th term of the AP = 32
• 12th term of the AP = 53

We know the general form of any term of an AP is given by,

• an = a + (n - 1)d

Here,

• an = nth term
• n = number of terms
• a = First term
• d = Common difference

Since 8th term of AP is 32, by using the general form of a term, we have:

• a + 7d = 32 _____(1.)

Also, 12th term of AP is 53, by using the general form of a term, we have:

• a + 11d = 53 _____(2.)

Subtracting equation (1) from equation (2)

⇒ a + 11d - (a + 7d) = 53 - 32

⇒ a + 11d - a - 7d = 21

⇒ 4d = 21

⇒ d = 21/4

Now substitute this value of d in equation (1),

➝ a + 7d = 32

➝ a + 7(21/4) = 32

➝ a + 147/4 = 32

➝ a = 32 - 147/4

➝ a = -19/4

Therefore, the first term of AP is -19/4 and it's common difference is 21/4.

Shortcut trick :-

Whenever we are given any two terms or AP, we can find the common difference by using the below equation.

$\underline{\boxed{\sf d = \dfrac{a_n - a_k}{n - k} }}$