Marie-Jeanne is experimenting with weights of 100 g Attached to a spring. She records the weight and length of the spring after attaching the weight. Number of 100 g weights attached to the spring 1 2 3 4 Length of spring [ cm ] 45 49 53 57 Write a general formula that represents the length of the spring if n weights are attached.

The length of the spring are in arithmetic progression 45 , 49 , 53 and 57 ⇒ a = 45 = First term and d = 57 - 53 = 4 = Common difference We know that, n th term of an A . P . is t n = a + n - 1 d Let L be the length of spring after n th weight ⇒ L = 45 + ( n - 1 ) × 4 ⇒ L = 45 + 4 n - 4 ⇒ L = 4 n + 41 ∴ The general formula of the length of spring if n weights are attached is L = 4 n + 41

Marie-Jeanne is experimenting with weights of 100 g Attached to a spring. She records the weight and length of the spring after attaching the weight. Number of 100 g weights attached to the spring 1 2 3 4 Length of spring [ cm ] 45 49 53 57 Calculate the supposed length of a spring if a 1 kg weight is attached.

The length of the spring are in Arithmetic progression 45 , 49 , 53 and 57 ⇒ a = 45 = First term and d = 57 - 53 = 4 = Common difference We know that, n th term can be written as, A . P . = t n = a + n - 1 d Let, L be the length of spring after n th weight ⇒ L = a + ( n - 1 ) d ⇒ L = 45 + ( n - 1 ) × 4 ⇒ L = 45 + 4 n - 4 ⇒ L = 4 n + 41 ∴ The General formula of the length of spring if n weights are attached = L n = 4 n + 41 The supposed length of a spring if a 1 kg weight is attached. n = 1 kg = 1000 gm = 10 × 100 gm ∴ 10 Spring attached, 100 gm each ⇒ L = 4 × 10 + 41 ⇒ L = 81 cm The length of spring if 1 kg is attached will be 81 c m .

Marie-Jeanne is experimenting with weights of 100 g Attached to a spring. She records the weight and length of the spring after attaching the weight. Number of 100 g weights attached to the spring 1 2 3 4 Length of spring [ cm ] 45 49 53 57 Write a general formula that represents the length of the spring if n weights are attached. Explain any limitations of Marie-Jeanne's experiment.

The length of the spring are in Arithmetic progression 45 , 49 , 53 and 57 ⇒ a = 45 = First term and d = 57 - 53 = 4 = Common difference We know that, n th term of an A . P . is t n = a + n - 1 d Let, L be the length of spring after n th weight ⇒ L = a + ( n - 1 ) d ⇒ L = 45 + ( n - 1 ) × 4 ⇒ L = 4 n + 41 As we know, when a weight is attached to spring it will expand and contract. Eventually the spring will hit the ground or the surface it is sitting on, so the length become constant and Also, the spring could break from too much weight. It is the limitation of marry-Jeanne experiment.

Marie-Jeanne is experimenting with weights of 100 g Attached to a spring. She records the weight and length of the spring after attaching the weight. Number of 100 g weights attached to the spring 1 2 3 4 Length of spring [ cm ] 45 49 53 57 If the spring stretches to 101 cm , what is the total weight that was attached to the spring?

The length of the spring are in Arithmetic progression 45 , 49 , 53 and 57 ⇒ a = 45 = First term and d = 57 - 53 = 4 = Common difference We know that, n th term of an A . P . is t n = a + n - 1 d Let, L be the length of spring after n th weight ⇒ L = a + ( n - 1 ) d ⇒ L = 45 + ( n - 1 ) × 4 ⇒ L = 45 + 4 n - 4 ⇒ L = 4 n + 41 If the spring stretches to 101 cm The total weight that was attached to the spring = n × 100 gm ⇒ 101 = 4 n + 41 ⇒ 101 - 41 = 4 n ⇒ 60 = 4 n ∴ n = 15 ∴ The total weight that was attached to the spring = n × 100 gm = 15 × 100 gm = 1500 gm ∴ If the spring stretches to 101 cm the total weight that was attached to the spring = 1500 gm = 1 . 5 kg

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Three consecutive terms of a geometric sequence are $x - 3, 6 and x + 2$ . Find all the possible values of $x$ .

Important Questions on From Patterns to Generalizations: Sequences and Series

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Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 11

Marie-Jeanne is experimenting with weights of $100 g$ Attached to a spring. She records the weight and length of the spring after attaching the weight.

Question Image

$\begin{array}{lcccc} \begin{array}{l} Number of 100 g weights \\ attached to the spring \end{array} & 1 & 2 & 3 & 4 \\ Length of spring [cm] & 45 & 49 & 53 & 57 \end{array}$

Write a general formula that represents the length of the spring if $n$ weights are attached.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 11

Marie-Jeanne is experimenting with weights of $100 g$ Attached to a spring. She records the weight and length of the spring after attaching the weight.

Question Image

$\begin{array}{lcccc} \begin{array}{l} Number of 100 g weights \\ attached to the spring \end{array} & 1 & 2 & 3 & 4 \\ Length of spring [cm] & 45 & 49 & 53 & 57 \end{array}$

Calculate the supposed length of a spring if a $1 kg$ weight is attached.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 11

Marie-Jeanne is experimenting with weights of $100 g$ Attached to a spring. She records the weight and length of the spring after attaching the weight.

Question Image

$\begin{array}{lcccc} \begin{array}{l} Number of 100 g weights \\ attached to the spring \end{array} & 1 & 2 & 3 & 4 \\ Length of spring [cm] & 45 & 49 & 53 & 57 \end{array}$

Write a general formula that represents the length of the spring if $n$ weights are attached. Explain any limitations of Marie-Jeanne's experiment.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 11

Marie-Jeanne is experimenting with weights of $100 g$ Attached to a spring. She records the weight and length of the spring after attaching the weight.

Question Image

$\begin{array}{lcccc} \begin{array}{l} Number of 100 g weights \\ attached to the spring \end{array} & 1 & 2 & 3 & 4 \\ Length of spring [cm] & 45 & 49 & 53 & 57 \end{array}$

If the spring stretches to $101 cm$ , what is the total weight that was attached to the spring?

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 12

Kostas gets a four-year bank loan to buy a new car that is priced at

€ 20987 .

After the four years, Kostas will have paid the bank a total of

€ 22960 .

What annual interest rate did the bank give him if the interest was compounded monthly?

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 13

The first seven numbers in row $14$ of Pascal's triangle are $1, 13, 78, 286, 715, 1287, 1716$ .

Complete the row and explain your strategy.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 13

The first seven numbers in row $14$ of Pascal's triangle are $1, 13, 78, 286, 715, 1287, 1716$ .

Complete the row and explain your strategy.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 1 - From Patterns to Generalizations: Sequences and Series>Chapter review>Q 14

Using the binomial theorem, expand

(3 x - y)^{6}

Three consecutive terms of a geometric sequence are x-3, 6 and x+2. Find all the possible values of x.

Important Questions on From Patterns to Generalizations: Sequences and Series

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Three consecutive terms of a geometric sequence are $x - 3, 6 and x + 2$ . Find all the possible values of $x$ .