Question

How many matches are in the 50th diagram of the pattern?

Answer 1

Answer: 101

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Step-by-step explanation:

n = diagram number

m = number of matches for diagram n

The given figures show that we have this info so far

`\begin{array}c \cline{1-2}n & m\\\cline{1-2}1 & 3\\\cline{1-2}2 & 5\\\cline{1-2}3 & 7\\\cline{1-2}\end{array}`

In other words, we have this sequence of values to represent the number of matches: 3, 5, 7

Each time we generate a new figure, we add 2 matches to the far right side. One match being horizontal and the other vertical.

This shows the common difference of this arithmetic sequence is d = 2.

The starting term is
`a_1 = 3`.

Let's find the nth term.

`a_n = a_1 + d(n-1)\\\\a_n = 3 + 2(n-1)\\\\a_n = 3 + 2n-2\\\\a_n = 2n+1\\\\`

Then we can determine the 50th term.

`a_n = 2n+1\\\\a_(50) = 2(50)+1\\\\a_(50) = 100+1\\\\a_(50) = 101\\\\`

There are 101 matches in the 50th diagram of the pattern.