Prove: $$\huge S_n = \frac{n(a_n + a_1)}2$$

Question

lgbasallote · Answer

it's an easy proof....but it's still nice to see some professional do it

lgbasallote · Answer

i wrote that formula wrong...

vishweshshrimali5 · Answer

See

$$\S_n = \cfrac{n}{2} (2a + (n-1)d)$$

lgbasallote · Answer

and..?

vishweshshrimali5 · Answer

Now
put

2a +(n-1)d = a + a+(n-1)d = a_1 + a_n

vishweshshrimali5 · Answer

You get ur answer !

lgbasallote · Answer

i don't think that proves anything...that was formula transformation

vishweshshrimali5 · Answer

see

a_n = a+(n-1)d

vishweshshrimali5 · Answer

That really does //////////// :)

vishweshshrimali5 · Answer

You don't want to use the first formula ?

lgbasallote · Answer

can you prove a_n = a + (n-1)d?

vishweshshrimali5 · Answer

I think yes

mathslover · Answer

yep

lgbasallote · Answer

i think you;re confused....this is proving...not substitution

shubhamsrg · Answer

suppose we have
1,2,3,4.....100
adding all, we can see it like this
(1+100) + (2+99) .... ( 50 + 51)

=> (101)*50 = (1 + 100) * (100/2)

you may generalize this..

vishweshshrimali5 · Answer

And u asked for a proof ;)

lgbasallote · Answer

that wasn't a proof @vishweshshrimali5 ....it was formula transformation

lgbasallote · Answer

and substitution isn't proving either.... @shubhamsrg

vishweshshrimali5 · Answer

Whatever........... leave it

I can prove that without using a_n formula

shubhamsrg · Answer

substitution ? o.O

vishweshshrimali5 · Answer

But you will have to accept that a_n = a+(n-1)d
That is basic def. of AP

vishweshshrimali5 · Answer

I am also surprised @shubhamsrg O.o

shubhamsrg · Answer

i am not! ;)

vishweshshrimali5 · Answer

;)

lgbasallote · Answer

are any of you familiar with mathematical induction?

vishweshshrimali5 · Answer

Yes

vishweshshrimali5 · Answer

You want that proof........

lgbasallote · Answer

what both of you did was substitution (which is NEVER accepted as proofs)

vishweshshrimali5 · Answer

Let P(n) be the given statement

lgbasallote · Answer

no. i want other proof

lgbasallote · Answer

induction is too dull and boring and predictable

shubhamsrg · Answer

you're amazing you know that ?

shubhamsrg · Answer

not you ! ;)

vishweshshrimali5 · Answer

@lgbasallote 
please give us a "hint" of the proof u want

lgbasallote · Answer

actually...it's surprising @vishweshshrimali5 does know induction...

shubhamsrg · Answer

hint? lol..

lgbasallote · Answer

any proof as long as it's valid and not induction

shubhamsrg · Answer

induction is one of the many great tools we have in mathematics today!

lgbasallote · Answer

preferrably, i like to see direct proof and contradiction

lgbasallote · Answer

i like deduction much more

vishweshshrimali5 · Answer

Ok lets try u want to deduce a formula from a continuous pattern

lgbasallote · Answer

time for the site to go down....

mathslover · Answer

bye

vishweshshrimali5 · Answer

bye

lgbasallote · Answer

so i suppose i'll just ask this again in the fture

vishweshshrimali5 · Answer

will check out later

mathslover · Answer

yep