Arithmetic Progression II

First observe the arithmetic proggresion. $123,103..$ continues by subtracting $20$ so the common difference is -20, the first term is |(123) so the equation is $n+(n-1)d$ in which d= difference between numbers so $123+(86-1)\times -20$ so now $123+(85)*-20$ $123+(-1700)$ so the answer is $\boxed{-1577}$

In A.P nth term is given by

n=a+(n-1)d

here n=86, a=123 & d=-20

=> n=123+(85)(-20)

=>n=-1577

therefore 86th term is -1577

a=123; d=-20; n=86. Thus a_{n} = a+(n-1)d = 123+(85)(-20) = 123 -1700 =-1577

easy way, the concept of sequence numbers with each - each different from 20. 123,103,83,63,43,23,3, ..... and so on by using the results of using the multiplication factor 3, ie 27, see the end of 7 that its contents

The formula used is:- tn = a + ( n - 1 ) * difference

Where tn = term number , a = first number of the sequence

In this sequence the numbers are in decreasing order. So, instead of difference 20 we should use -20.

Tn = a + ( n - 1 ) * difference

= 123 - 20n + 20

= -20n + 143

So we got the formula and have to find the 86th term.

= -20n + 143

= ( -20 * 86 ) + 143

= -1720 + 143

= -1577