Arithmetic Progression I

First observe the arithmetic proggresion. $-3,5,13..$ continues bu adding $8$ so the common difference is 8, the first term is |(-3) so the equation is $a+(n-1)d$ in which d= difference between numbers so $-3+(65-1)\times 8$ so now $-3+(64)*8$ $-3+(512)$ so the answer is $\boxed{509}$

$Here\quad \\ \qquad { a }_{ 1 }=-3\quad ,d=5-(-3)=13-5=\quad 8\\ Using\quad this\quad formula\\ \qquad \boxed { { a }_{ n }={ a }_{ 1 }+(n-1)d } \\ \qquad { a }_{ 65 }=\quad -3+(65-1)(8)\\ \qquad \quad \quad =\quad -3+(64)(8)\\ \qquad \quad \quad =\quad -3+512\\ Thus\quad the\quad answer\quad is\quad \\ \qquad \qquad \boxed { { a }_{ 65 }=509 }$

an = a1 + (n - 1)r

65 = -3 + 64 . 8 = 509

The formula used is:- tn = a + ( n - 1 ) * difference Where tn = term number , a = first number of the sequence

Tn = a + ( n - 1 ) * difference

  = -3 + ( n - 1 ) * 8

  = -3 + 8n -8

  = 8n - 11

So we have got the formula and have to find the 65th term.

= 8n - 11

= ( 8 * 65 ) - 11

= 520 - 11

= 509