Question

simplify:

$1+3+7+15+31+...$ to $n$ terms.

Answer 1

$S=1+3+7+15+31+.....n$$terms$S=1+\left( { 2 }^{ 2 }-1 \right) +\left( { 2 }^{ 3 }-1 \right) +\left( { 2 }^{ 4 }-1 \right) +\left( { 2 }^{ 5 }-1 \right) +...{ 2 }^{ n }-1S=1-(n-1)+{ 2 }^{ 2 }+{ 2 }^{ 3 }+{ 2 }^{ 4 }+...{ 2 }^{ n }n-1$terms$S=1-(n-1)+{ 2 }^{ 2 }\cfrac { \left( 1-{ 2 }^{ n-1 } \right) }{ 1-2 } S=1-n+1+4\left( { 2 }^{ n-1 }-1 \right) S=1-n+1+{ 2 }^{ 2 }\left( { 2 }^{ n-1 }-1 \right) S=1-n+1+{ 2 }^{ n+1 }-4S={ 2 }^{ n+1 }-n-2$