Question

find the sum of 'n' numbers of the following A.P.
a) 1,4,7 (55 terms)
c) 23,21,19 (45 terms)
f)4,8,12 (90 terms)

Answer 1

Gauss's Method

The nth term of each arithmetic progression is $3n-2$, $-2n+25$, $4n$.

We could get a hint from Gauss's method.

 

Natural Numbers

Let us denote $S_n$ as the sum of the first n terms.

• $S_{100} = 1 + 2 + 3 + ... + 98 + 99 + 100$
• $S_{100} = 100 + 99 + 98 + ... + 3 + 2 + 1$

It's nothing more or less than a commutative property. This is possible because the number of terms is finite.

Let us sum the two equations.

$\implies 2S_{100} = 10 1+ 101 + 101 + ... + 101 + 101 + 101$

$\implies 2S_{100} = 101 \times 100$

$\therefore S_{100} = 5050$

 

Application

So, let's apply this to our arithmetic progressions.

First things first. Let the first and last term be $a$ and $l$, the common difference be $d$.

• $S_{n} = a + ( a + d ) + ... + ( l - d ) + l$
• $S_{n} = l + ( l - d ) + ... + ( a + d ) + a$

$\implies 2S_{n}=(a+l)+(a+l)+...+(a+l)+(a+l)$

$\implies 2S_{n} = n ( a + l )$

$\implies S_{n} = \dfrac{ n ( a + l ) }{ 2 }$

 

Now let's solve each question.

55th term of the following A.P is 163.

$S_{55}=\dfrac{55(1+163)}{2} = 4510$

45th term of the following A.P is -65.

$S_{45} = \dfrac{45 ( 23 - 65 ) }{2} = -945$

90th term of the following A.P is 360.

$S_{90} = \dfrac{ 90 ( 4 + 360 ) }{2} = 16380$

 

Learn more

About harmonic progression

• 1, 2, 3, 4, ...

This is a normal A.P. But

• 1, $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4}$, ...

This is the inverse of the A.P, which case is called the harmonic progression.

What will be the value if we add all terms until infinity?

$1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} + ...$

$\geq 1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + ...$

$= 1 + \dfrac{1}{2} + \dfrac{2}{4} + \dfrac{4}{8} + ... = 1 + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + ...$

So, the sum goes to infinity.

 

Information from

• Harmonic Series (W.pedia)
• Mathologer (Utube)

You can refer to

• The Leaning Tower of Lire (W.pedia)
• Clever Carl (nrich)