A sequence is deﬁned recursively as follows: $$a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$$
It is proposed that an explicit formula for this sequence is $$a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }$$
Use mathematical induction to check whether this proposed formula is correct.

Question

A sequence is deﬁned recursively as follows: $$a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$$ 
It is proposed that an explicit formula for this sequence is $$a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }$$ 
Use mathematical induction to check whether this proposed formula is correct.

Quizplus · Accepted Answer

The Answer of A sequence is deﬁned recursively as follows: $$a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$$ 
It is proposed that an explicit formula for this sequence is $$a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }$$ 
Use mathematical induction to check whether this proposed formula is correct.

A Sequence Is Deﬁned Recursively as Follows a0=2 and ak=4ak−1+1 for all k≥1.a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .a0​=2 and ak​=4ak−1​+1 for all k≥1.

A Sequence Is Deﬁned Recursively as Follows $a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$