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Use mathematical induction to prove the given statement for all positive integers [tex]\( n \)[/tex].

[tex]\[ 3 + 11 + 19 + \ldots + (8n - 5) = n(4n - 1) \][/tex]

Part: [tex]\(0 / 6\)[/tex]

Part 1 of 6

Let [tex]\( P_n \)[/tex] be the statement:

[tex]\[ 3 + 11 + 19 + \ldots + (8n - 5) = n(4n - 1) \][/tex]

Show that [tex]\( P_n \)[/tex] is true for [tex]\( n = \square \)[/tex].

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Note: [tex]\( \square \)[/tex] indicates the need for an input or a box to fill in the specific value of [tex]\( n \)[/tex] (typically [tex]\( n = 1 \)[/tex] for the base case).



Answer :

GinnyAnswer
Part 1 of 6

Let [tex]\( P_n \)[/tex] be the statement: [tex]\( 3 + 11 + 19 + \ldots + (8n - 5) = n(4n - 1) \)[/tex].

We need to show that [tex]\( P_n \)[/tex] is true for [tex]\( n = 1 \)[/tex].

When [tex]\( n = 1 \)[/tex]:

Left-hand side (LHS):
[tex]\[ 3 + 11 + 19 + \ldots + (8 \cdot 1 - 5) = 8 \cdot 1 - 5 = 3 \][/tex]

Right-hand side (RHS):
[tex]\[ 1(4 \cdot 1 - 1) = 1(4 - 1) = 1 \cdot 3 = 3 \][/tex]

Thus, the statement [tex]\( P_1 \)[/tex] is:
[tex]\[ 3 = 3 \][/tex]

Since both sides match, [tex]\( P_1 \)[/tex] is true.