Question

find the first term of A.P. if the sum of three consecutive term is 21
​

Answer 1

Answer:

The first term is 7.

Step-by-step explanation:

Let us assume the three consecutive numbers to be '(a - d)', 'a' and '(a + d)'.

[Explanation for why we've chosen the above terms given at the end]

Given that:

Sum of consecutive terms is 21.

That is,

$\Longrightarrow \tt a-d+a+a+d=21$

$\Longrightarrow \tt a+a+a=21$

$\Longrightarrow \tt 3a=21$

$\Longrightarrow \tt a= \dfrac{21}{3}$

$\Longrightarrow \tt a=7$

Therefore, the First term of the AP is 7.

Why do we take a - d, a, and a + d as the consecutive terms?

It's because if a term in an AP can be expressed as 'a', the preceeding term can be expressed as 'a' subtracted by the common difference 'd', and the suceeding term as 'a' added to the common difference. 3 such consecutive terms can be expressed as 'a - d', 'a', and 'a + d'

Example, Take three consecutive terms, 3, 4 and 5.

4, the middle term can be written as ⇒ a

3, the first/preceeding term can be written as ⇒ a - 1 (4 - 1 = 3)

5, the third/succeeding term can be written as ⇒ a + 1 (4 + 1 = 5)

Other types of Selection of terms.

For four consecutive terms:

$\tt \Longrightarrow (a-3d), (a-d), (a+d), (a+3d)$

For five consecutive terms:

$\tt \Longrightarrow (a-2d), (a-d), (a), (a+d), (a+2d)$

Answer 2

$\bf{\Huge{\underline{\boxed{\bf{\red{ANSWER\::}}}}}}$

$\bf{\Large{\underline{\bf{Given\::}}}}$

If the sum of three consecutive term is 21.

$\bf{\Large{\underline{\bf{To\:find\::}}}}$

The first term of Arithmetic progression.

$\bf{\Large{\underline{\bf{\blue{Explanation\::}}}}}$

Let the three consecutive term is;

• First term = (a-d)
• Second term = a
• Third term = (a+d)

According to the question:

→ (a-d) + a + (a+d) = 21

→ $\bf{a\cancel{-d}+a+a\cancel{+d}=21}$

→ 3a = 21

→ a = $\bf{\cancel{\frac{21}{3}}}$

→ a = 7

Thus,

The first term of an A.P. is 7.