Gr12 Mathematics: Ch 1 Sum Number Patterns, Sequences and Series

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Questions and Answers

What is the formula to find the nth term of an arithmetic sequence?

  • Tn = a + (n - 1)d (correct)
  • Tn = a - (n - 1)d
  • Tn = an + d
  • Tn = an - d

How do you determine if a sequence is arithmetic?

  • By checking if the product of consecutive terms is constant
  • By checking if the sum of consecutive terms is constant
  • By verifying if the ratio of consecutive terms is constant
  • By verifying if the difference between consecutive terms is constant (correct)

What is the graphical representation of an arithmetic sequence?

  • A circle
  • A parabola
  • A curved line
  • A straight line (correct)

What is the arithmetic mean of two numbers?

<p>The average of the two numbers (B)</p> Signup and view all the answers

What is the characteristic of an arithmetic sequence when d is positive?

<p>The sequence increases (B)</p> Signup and view all the answers

How do you find the common difference of an arithmetic sequence?

<p>By subtracting consecutive terms (A)</p> Signup and view all the answers

What is the purpose of the formula Tn = a + (n - 1)d?

<p>To find the nth term of an arithmetic sequence (C)</p> Signup and view all the answers

What is the pattern of the terms in an arithmetic sequence?

<p>Linear (C)</p> Signup and view all the answers

What is the formula for the nth term of a geometric sequence?

<p>Tn = ar^(n-1) (D)</p> Signup and view all the answers

What is the geometric mean between two numbers a and b?

<p>√ab (B)</p> Signup and view all the answers

What is the characteristic of a geometric sequence when the common ratio r is negative?

<p>The terms alternate in sign (B)</p> Signup and view all the answers

How do you calculate the common ratio r of a geometric sequence?

<p>r = Tn / T(n-1) (A)</p> Signup and view all the answers

What is a finite series?

<p>The sum of a specific number of terms of a sequence (C)</p> Signup and view all the answers

What is the notation used to represent the sum of terms in a sequence?

<p>Sigma notation (D)</p> Signup and view all the answers

What is the general form of sigma notation?

<p>∑(i=m)^(n) T_i = T_m + T_{m+1} + ... + T_n (C)</p> Signup and view all the answers

What is an infinite series?

<p>The sum of infinitely many terms of a sequence (D)</p> Signup and view all the answers

What is the graphical representation of a geometric sequence?

<p>An exponential graph (B)</p> Signup and view all the answers

How do you test if a sequence is geometric?

<p>Calculate the ratio between consecutive terms (D)</p> Signup and view all the answers

What is the formula to find the sum of the first n terms of a geometric sequence?

<p>$S_n = \frac{a(1 - r^n)}{1 - r}$ (B)</p> Signup and view all the answers

What is the common ratio in a geometric sequence?

<p>The constant value multiplied by the previous term to get the next term (A)</p> Signup and view all the answers

What is the condition for an infinite geometric series to be convergent?

<p>$-1 &lt; r &lt; 1$ (A)</p> Signup and view all the answers

What is the formula for the sum of an infinite geometric series?

<p>$S_\infty = \frac{a}{1 - r}$ (B)</p> Signup and view all the answers

What is the common difference in an arithmetic sequence?

<p>The difference between each term and the previous term (D)</p> Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence?

<p>$T_n = a + (n - 1)d$ (D)</p> Signup and view all the answers

What is the sum of the first 100 terms of an arithmetic sequence?

<p>5050 (B)</p> Signup and view all the answers

What is the method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

<p>Pairing the numbers in ascending and descending order and adding the corresponding pairs (D)</p> Signup and view all the answers

What is the sum of the first n terms of a finite geometric series with first term a and common ratio r?

<p>$S_n = \frac{a(1 - r^n)}{1 - r}$ (C)</p> Signup and view all the answers

What is the correct formula for the sum of a finite arithmetic series if the last term is unknown?

<p>S_n = (n / 2)(2a + (n - 1)d) (B)</p> Signup and view all the answers

What is the purpose of deriving the general formula for a finite arithmetic series?

<p>To sum the arithmetic sequence from the first term to the last term (D)</p> Signup and view all the answers

What is the sum of an infinite geometric series with first term a and common ratio r?

<p>$S_\infty = \frac{a}{1 - r}$ (C)</p> Signup and view all the answers

What is the correct expansion of the sum of a finite arithmetic series?

<p>S_n = a + (a + d) + (a + 2d) + ... + (l - 2d) + (l - d) + l (C)</p> Signup and view all the answers

What is the condition for an infinite geometric series to be divergent?

<p>$r &gt; 1$ or $r &lt; -1$ (B)</p> Signup and view all the answers

What is the correct formula for the sum of a finite arithmetic series when the first and last terms are known?

<p>S_n = (n / 2)(a + l) (B)</p> Signup and view all the answers

What is the general formula to find the nth term of an arithmetic sequence?

<p>Tn = a + (n - 1)d (B)</p> Signup and view all the answers

Which of the following is a characteristic of an arithmetic sequence?

<p>The sequence forms a linear pattern when plotted on a graph (C)</p> Signup and view all the answers

What can be calculated by finding the average of two numbers to form an arithmetic sequence?

<p>Arithmetic mean (C)</p> Signup and view all the answers

What is the formula to find the nth term of a geometric sequence?

<p>$T_n = ar^{n-1}$ (A)</p> Signup and view all the answers

How do you find the number of terms in an arithmetic sequence?

<p>Set the nth term equal to a specific value and solve for n (C)</p> Signup and view all the answers

What is the purpose of calculating the differences between consecutive terms in an arithmetic sequence?

<p>To find the common difference (B)</p> Signup and view all the answers

What is the method to check if a sequence is geometric?

<p>Find the ratio between consecutive terms (C)</p> Signup and view all the answers

What is the formula for the sum of the first n terms of a finite arithmetic series when the last term is unknown?

<p>$S_n = rac{n}{2} (2a + (n - 1) d)$ (C)</p> Signup and view all the answers

What is the characteristic of a geometric sequence when 0 < r < 1?

<p>The sequence decays exponentially (A)</p> Signup and view all the answers

Why do we need to derive the general formula for a finite arithmetic series?

<p>To calculate the sum of the first n terms of an arithmetic sequence. (B)</p> Signup and view all the answers

What is true about the terms of an arithmetic sequence?

<p>The terms form a linear pattern when plotted on a graph (A)</p> Signup and view all the answers

What is the general formula for the sum of a finite arithmetic series?

<p>$S_n = rac{n}{2} (a + l)$ (C)</p> Signup and view all the answers

What does the sum of the terms of a sequence represent?

<p>A series (C)</p> Signup and view all the answers

How do you identify the first term of an arithmetic sequence?

<p>It is given as a (C)</p> Signup and view all the answers

What is the result of plotting the nth term against n in an arithmetic sequence?

<p>A straight line (C)</p> Signup and view all the answers

What is the notation used to represent the sum of terms in a sequence?

<p>Sigma notation (B)</p> Signup and view all the answers

How do we derive the general formula for a finite arithmetic series?

<p>By summing the arithmetic sequence from the first term to the last term. (B)</p> Signup and view all the answers

What is an infinite series?

<p>A sum of infinitely many terms in a sequence (D)</p> Signup and view all the answers

What is the formula for the sum of the first n terms of a finite arithmetic series when the first and last terms are known?

<p>$S_n = rac{n}{2} (a + l)$ (B)</p> Signup and view all the answers

What is the general form of sigma notation?

<p>$\sum_{i=m}^{n} T_i = T_m + T_{m+1} + \cdots + T_{n-1} + T_n$ (B)</p> Signup and view all the answers

What is the purpose of finding the common ratio in a geometric sequence?

<p>To determine if the sequence is geometric (D)</p> Signup and view all the answers

What is the result of plotting T_n against n for a geometric sequence?

<p>An exponential graph (A)</p> Signup and view all the answers

What can be used to find any term in a geometric sequence?

<p>The formula T_n = ar^(n-1) (C)</p> Signup and view all the answers

What is the condition for an infinite geometric series to be convergent?

<p>-1 &lt; r &lt; 1 (B)</p> Signup and view all the answers

What is the formula for the sum of a finite geometric series?

<p>a(1 - r^n) / (r - 1) (D)</p> Signup and view all the answers

What is the definition of a finite arithmetic series?

<p>A series with a fixed number of terms (B)</p> Signup and view all the answers

What is the formula for the sum of an infinite geometric series?

<p>a / (1 - r) (D)</p> Signup and view all the answers

What is the definition of an infinite geometric series?

<p>A series where the number of terms goes to infinity (A)</p> Signup and view all the answers

What is the purpose of deriving the general formula for a finite geometric series?

<p>To find the sum of a finite geometric series (B)</p> Signup and view all the answers

What is the formula for the nth term of a geometric sequence?

<p>ar^(n - 1) (D)</p> Signup and view all the answers

What is the definition of a geometric sequence?

<p>A sequence where each term after the first is found by multiplying by a constant value (B)</p> Signup and view all the answers

What is the sum of the first n terms of a finite arithmetic series?

<p>n(a + l) / 2 (D)</p> Signup and view all the answers

What is Karl Friedrich Gauss's method for finding the sum of the first 100 integers?

<p>Writing the numbers in ascending and descending order and adding corresponding terms (C)</p> Signup and view all the answers

If an arithmetic sequence has a first term of 5 and a common difference of 2, what is the value of the 10th term?

<p>23 (B)</p> Signup and view all the answers

What is the common difference of an arithmetic sequence where the 5th term is 11 and the 8th term is 17?

<p>2 (C)</p> Signup and view all the answers

If an arithmetic sequence has a first term of 3 and a common difference of -2, what is the value of the 7th term?

<p>-11 (D)</p> Signup and view all the answers

What is the number of terms in an arithmetic sequence with a first term of 2, a common difference of 3, and a last term of 50?

<p>17 (C)</p> Signup and view all the answers

If two terms of an arithmetic sequence are 7 and 13, what is the common difference?

<p>3 (B)</p> Signup and view all the answers

What is the arithmetic mean between 5 and 15?

<p>10 (B)</p> Signup and view all the answers

If an arithmetic sequence has a common difference of 4, what is the difference between the 10th term and the 5th term?

<p>20 (B)</p> Signup and view all the answers

What is the graph of an arithmetic sequence when plotted on a coordinate plane?

<p>A straight line (A)</p> Signup and view all the answers

What is the value of the common ratio r in a geometric sequence where the second term is 5 and the fourth term is 45?

<p>3/5 (B)</p> Signup and view all the answers

What is the formula for the sum of the first n terms of a finite geometric series with first term a and common ratio r?

<p>a(1 - r^n)/(1 - r) (A)</p> Signup and view all the answers

What is the graphical representation of a geometric sequence?

<p>An exponential curve (A)</p> Signup and view all the answers

What is the condition for an infinite geometric series to be convergent?

<p>r &lt; 1 (A)</p> Signup and view all the answers

What is the formula for the nth term of a geometric sequence?

<p>ar^(n-1) (B)</p> Signup and view all the answers

What is the sum of an infinite geometric series with first term a and common ratio r?

<p>a/(1 - r) (D)</p> Signup and view all the answers

What is the characteristic of a geometric sequence when r is negative?

<p>Alternating signs (B)</p> Signup and view all the answers

What is the purpose of sigma notation?

<p>To represent the sum of terms in a sequence (A)</p> Signup and view all the answers

What is the difference between a finite series and an infinite series?

<p>A finite series has a fixed number of terms, while an infinite series has an infinite number of terms (C)</p> Signup and view all the answers

What is the formula for the geometric mean between two numbers a and b?

<p>√(ab) (D)</p> Signup and view all the answers

Which of the following is a necessary condition for an infinite geometric series to be convergent?

<p>-1 &lt; r &lt; 1 (B)</p> Signup and view all the answers

What is the formula for the sum of a finite geometric series with first term a and common ratio r?

<p>a(1 - r^n)/(1 - r) (C)</p> Signup and view all the answers

What is the sum of the first 100 integers?

<p>5050 (D)</p> Signup and view all the answers

What is the common ratio of a geometric sequence where each term is twice the previous term?

<p>2 (D)</p> Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence with first term a and common difference d?

<p>a + (n - 1)d (D)</p> Signup and view all the answers

What is the sum of an infinite geometric series with first term a and common ratio r?

<p>a/(1 - r) (A)</p> Signup and view all the answers

What is the condition for an infinite geometric series to be divergent?

<p>r &gt; 1 or r &lt; -1 (D)</p> Signup and view all the answers

What is the purpose of deriving the general formula for a finite arithmetic series?

<p>To calculate the sum of a finite series (B)</p> Signup and view all the answers

What is the characteristic of an arithmetic sequence when the common difference d is positive?

<p>The terms increase by a constant value (A)</p> Signup and view all the answers

What is the result of plotting the nth term against n in an arithmetic sequence?

<p>A straight line (C)</p> Signup and view all the answers

If the first term of an arithmetic sequence is 2 and the common difference is 3, what is the sum of the first 10 terms?

<p>155 (B)</p> Signup and view all the answers

In an arithmetic sequence, if the last term is known to be 25 and the common difference is 2, what is the formula to find the sum of the first n terms?

<p>S_n = (n/2) (a + l) (C)</p> Signup and view all the answers

What is the sum of the first 20 terms of an arithmetic sequence with a first term of 5 and a common difference of 4?

<p>420 (B)</p> Signup and view all the answers

If the sum of the first 15 terms of an arithmetic sequence is 645, what is the formula to find the first term of the sequence?

<p>a = (2S_n) / n - (n-1)d (D)</p> Signup and view all the answers

What is the general formula for the sum of the first n terms of an arithmetic sequence if the last term is unknown?

<p>S_n = (n/2) [2a + (n-1)d)] (C)</p> Signup and view all the answers

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