samaantar shredhi

ganit mein samaantar shredhi (angreji: Arithmetic progression) athva samaantar anukram sankhyaaon ka ek aisa anukram hai jiske do kramaagat pado ka antar niyat hota hai. jaise anukram 4, 7, 10, 13, 16 ... saarv antar 3 ke saath ek samaantar shredhi hai.

yadi kisi samaantar shredhi ka pratham pad ${\displaystyle a_{1}}$ aur saarv antar d hai to shredhi ka ivaaain pad (${\displaystyle a_{i}}$) nimn prakaar likha jaata hai

${\displaystyle a_{i}=id+a_{0}\,}$

aur vyaapak roop mein

${\displaystyle \ a_{n}=a_{m}+(n-m)d.}$

samaantar shredhi ke parimit bhaag ko parimit samaantar shredhi kehte hain aur saamaanyat: keval samaantar shredhi kaha jaata hai. parimit samaantar shredhi ka yog ko samaantar shreni kehte hain.

samaantar shredhi ka vyavahaar iske saarv antar d par nirbhar karta hai. yadi saarv antar:

• dhanaatmak hai to iske pad dhanaatmak anant ki or agrasar honge.
• rinaatmak hai to iske pad rinaatmak anant ki or agrasar honge.

samaantar shredhi ke n padon ka yog

${\displaystyle s_{n}=a_{0}+(a_{0}+d)+(a_{0}+2d)+\dotsb +(a_{0}+(n-1)d)+(a_{0}+nd)=\sum _{i=0}^{n}(a_{0}+id)}$.

${\displaystyle s_{n}=\sum _{i=0}^{n}(i\cdot d+a_{0})=a_{0}(n+1)+d\,{\frac {n(n+1)}{2}}=(n+1)\left(a_{0}+d\,{\frac {n}{2}}\right)}$.

${\displaystyle \sum _{k=1}^{n}k=1+2+3+\dotsb +n={\frac {n(n+1)}{2}}}$

${\displaystyle \sum _{k=1}^{n}(2k-1)=\sum _{k=0}^{n-1}(2k+1)=1+3+5+7+\dotsb +(2n-1)=n^{2}}$

yahaaain ${\displaystyle a_{0}=1\,}$, ${\displaystyle d=2\,}$.

• gunottar shredhi
• haraatmak shredhi

• Arithmetic Series bei MathWorld (englisch)