Complete the table and find the general rule for the pattern: 3; 5; 7; 9...

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Here are the solutions to the problems. 3. Complete the table below The pattern for the "Number of matchsticks" is an arithmetic sequence where each term increases by 2. The sequence starts with 3, 5, 7, 9, 11, 13, 15... The missing values in the table are for Stage number 8, 9, and 10. For Stage number 8: 15 + 2 = 17; For Stage number 9: 17 + 2 = 19; For Stage number 10: 19 + 2 = 21 The completed table values for the number of matchsticks are: Stage 8: 17 Stage 9: 19 Stage 10: 21 4. Study the numeric pattern below and answer the questions that follow. The given numeric pattern is 3; 5; 7; 9... This is an arithmetic sequence with the first term a = 3 and a common difference d = 2. a) Extend the pattern by the next three terms. Step 1: Identify the common difference. d = 5 − 3 = 2 Step 2: Calculate the next three terms after 9 by adding the common difference. 9 + 2 = 11; 11 + 2 = 13; 13 + 2 = 15 The next three terms are 11, 13, 15. b) Determine the general rule for the pattern. Step 1: Identify the first term (a) and common difference (d). The first term is a = 3. The common difference is d = 2. Step 2: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n−1)d. T_n = 3 + (n−1)2 Step 3: Simplify the expression. T_n = 3 + 2n − 2; T_n = 2n + 1; The general rule for the pattern is T_n = 2n + 1. c) Determine the 12th term in the pattern. Step 1: Use the general rule T_n = 2n + 1. We need to find the 12th term, so substitute n = 12. Step 2: Calculate the 12th term. T₁₂ = 2(12) + 1; T₁₂ = 24 + 1; T₁₂ = 25 The 12th term in the pattern is 25. d) Which position of the term in the pattern will yield to 31? Step 1: Set the general rule T_n = 2n + 1 equal to 31. 2n + 1 = 31 Step 2: Solve for n. 2n = 31 − 1; 2n = 30; n = 30/2; n = 15 The position of the term that yields 31 is the 15th term.

Complete the table and find the general rule for the pattern: 3; 5; 7; 9...

Answers

Step-by-step solution

Part a· 2 steps

Identify the common difference.

Calculate the next three terms after 9 by adding the common difference.

Part b· 3 steps
$T_n = 2n + 1$

Identify the first term ( $a$ ) and common difference ( $d$ ).

Use the formula for the $n$ -th term of an arithmetic sequence, $T_n = a + (n-1)d$ .

Simplify the expression.

Part c· 2 steps25

Use the general rule $T_n = 2n + 1$ .

Calculate the 12th term.

Part d· 2 steps
$15\text{th}$

Set the general rule $T_n = 2n + 1$ equal to 31.

Solve for $n$ .

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Complete the table and find the general rule for the pattern: 3; 5; 7; 9...

Answers

Step-by-step solution

Part a· 2 steps

Identify the common difference.

Calculate the next three terms after 9 by adding the common difference.

Part b· 3 stepsTn=2n+1T_n = 2n + 1Tn​=2n+1

Identify the first term (aaa) and common difference (ddd).

Use the formula for the nnn-th term of an arithmetic sequence, Tn=a+(n−1)dT_n = a + (n-1)dTn​=a+(n−1)d.

Simplify the expression.

Part c· 2 steps25

Use the general rule Tn=2n+1T_n = 2n + 1Tn​=2n+1.

Calculate the 12th term.

Part d· 2 steps15th15\text{th}15th

Set the general rule Tn=2n+1T_n = 2n + 1Tn​=2n+1 equal to 31.

Solve for nnn.

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Part b· 3 steps
$T_n = 2n + 1$

Identify the first term ( $a$ ) and common difference ( $d$ ).

Use the formula for the $n$ -th term of an arithmetic sequence, $T_n = a + (n-1)d$ .

Use the general rule $T_n = 2n + 1$ .

Part d· 2 steps
$15\text{th}$

Set the general rule $T_n = 2n + 1$ equal to 31.

Solve for $n$ .