An algebra problem by Hammad Shah

Algebra Level 2

How many terms of the series -7+(-5)+(-3)+...... amount to 65?

9 12 11 13

Hammad Shah by Hammad Shah , 25, Pakistan

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2 solutions

Ashley Shamidha
Sep 12, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Mahdi Raza
Jun 29, 2020

( − 7 ) + ( − 5 ) + ( − 3 ) + ( − 1 ) + 1 + 3 + … ( 2 n + 1 ) = 65 1 + 3 + … ( 2 n + 1 ) = 65 − ( − 16 ) 1 + 3 + … ( 2 n + 1 ) = 81 n 2 = 81 Sum of odd numbers = n 2 n = 9 \begin{aligned} (-7) + (-5) + (-3) + (-1) + 1 + 3 + \ldots (2n + 1) &= 65 \\ 1 + 3 + \ldots (2n + 1) &= 65- (-16) \\ 1 + 3 + \ldots (2n + 1) &= 81 \\ n^2 &= 81 &\blue{\text{Sum of odd numbers} = n^2} \\ n &= 9 \end{aligned}

Total numbers in this sequence are 4 negative and 9 positive numbers = 13 \boxed{13}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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