Question

Please help meeeeeeeee

Answer 1

`\textsf{2.2.1)} \quad \sf True`

`\textsf{2.2.2)} \quad T_n=3n+2`

Explanation:

A numeric pattern is a sequence of numbers that follows a consistent rule or pattern, where each term is related to the previous term in a specific way. These patterns can involve simple operations like addition or multiplication, and they help us understand how numbers behave and make predictions.

From observation of the given diagram, the number of black dots in each figure is:

• Figure 1: 5 black dots
• Figure 2: 8 black dots
• Figure 3: 11 black dots

Each figure has 3 more black dots than the previous figure. Therefore, the pattern is an example of a numeric pattern since the common difference between terms is 3.

As there is a common difference between terms, the sequence is arithmetic. Therefore, to determine the nth term for the number of black dots, we can use the general formula for an arithmetic sequence:

`\boxed{\begin{minipage}{8 cm}\underline{General form of an arithmetic sequence}\\\\$T_n=a+(n-1)d$\\\\where:\\\phantom{ww}$\bullet$ $T_n$ is the nth term. \\ \phantom{ww}$\bullet$ $a$ is the first term.\\\phantom{ww}$\bullet$ $d$ is the common difference between terms.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}`

Given that the first term is a = 5 and the common difference is d = 3, substitute these values into the formula and simplify:

`T_n=5+(n-1)3`

`T_n=5+3n-3`

`T_n=3n+2`

Therefore, the equation for the nth term for the number of black dots is:

`\boxed{T_n=3n+2}`