Question

For an A.P., the first term is 1 and the last term is 20. The sum of
all the terms is 420. What is value of n?
(A) 21
(B) 40 (C) 20 (D) 42​

Answer 1

Step-by-step explanation:

First term ( a ) = 1

Last term ( l ) = 20.

Sum of 'N' terms = 420

Sum of 'N' terms of an A.P. = N/2 ( a + l )

420 = N/2 ( 1 + 20 )

420 x 2 = N ( 21 )

840 = 21 x N

N = 840 / 21

N = 40

So, there are 40 terms in that A.P.

Like my answer if you find it useful!

Answer 2

Answer

Choice B: The value of n is 40.

Given conditions

• $a=1$ (The 1st term)
• $l=20$ (The last term)
• $S_{n}=420$ (Sum of the terms)

Basis

Gauss's Method

Let the sum of all terms until the last n-th term be $S_n$.

And let the first term be $a$, the last term $l$, and the common difference $d$.

Then [1]

• $S_n= a+(a+d)+(a+2d)+...+(l-2d)+(l-d)+l$

• $S_n= l+(l-d)+(l-2d)+...+(a+2d)+(a+d)+a$

$\Longleftrightarrow 2S_n= (a+l)+(a+l)+(a+l)+...+(a+l)+(a+l)+(a+l)$

$\therefore S_n=\dfrac{n(a+l)}{2}$

Hence we can find the sum of the series using the first and last term. [2]

Application

Given that $a=1$ and $l=20$,

$S_n=\dfrac{n(1+20)}{2}$

$=\dfrac{21n}{2}$, which is 420.

Hence the value of n is 40.

More information

[1] We can find the reversed series. Then we add the reversed series by the previous one. And then we derive the Gauss formula.

[2] We can find it using the common difference, by the n-th term of the A.P.

The n-th term is last, then $a_n=a+(n-1)d$ is $l$.

Then we get

$S_{n}=\dfrac{n\{a+(n-1)d\}}{2}$