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Sequences and Series Test 10

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Sequences and Series Test 10
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  • Question 1
    1 / -0
    Find the next number in the series.
    $$3, 6, 9, 12, 15,....$$
    Solution
    The next number is $$18$$.
    Since the numbers are multiple of $$3$$.
    $$3, 6, 9, 12, 15, \underline {18 }$$.
  • Question 2
    1 / -0
    The area of triangle whose vertices are $$A (-3, -1), B(5, 3)$$ and $$C(2, -8)$$ is ____ $$\text{ sq. units}$$.
    Solution
    We know that the area of the triangle whose vertices are $$\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),$$ and $$(x_{3},y_{3})$$ is $$\cfrac{1}{2}\left | x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}) +x_{3}(y_{1}-y_{2})\right |$$

    The given vertices of the triangle are $$A(-3,-1), B(5,3)$$ and $$C(2,-8)$$.
    So, by using the above formula,
    $$\begin{aligned}{}\text{Area of the triangle} &= \frac{1}{2} {[ - 3(3 - ( - 8)) + 5( - 8 - ( - 1)) + 2( - 1 - 3)]}\\ &= \frac{1}{2}{[ - 33 - 35 - 8]}\\ &= \frac{{[ - 76]}}{2}\\& = \frac{{76}}{2}\quad\quad\quad\quad\dots[\text{Area can never be negative so, we ignore negative sign}]\\& = 38\text{ sq. units}\end{aligned}$$

    So, the area of the triangle is equal to $$38\text{ sq. units}$$.
  • Question 3
    1 / -0
    Given three vertices of a triangle whose coordinates are A (1, 1), B (3, -3) and (5, -3). Find the area of the triangle.
    Solution
    We know that the area of the triangle whose vertices are $$\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),$$ and $$(x_{3},y_{3})$$ is $$\cfrac{1}{2}\left | x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}) +x_{3}(y_{1}-y_{2})\right |$$
    The three vertices of the triangle are $$A(1,1), B(3,-3), C(5,-3)$$
    Area of triangle $$=\dfrac{|1(-3-(-3)+3(-3-1)+5(1-(-3))|}{2}$$
    $$=\dfrac{|1(0)+3(-4)+4(4)|}{2}$$
    $$=\dfrac{|-12+20|}{2}$$
    $$=\dfrac{8}{2}$$
    $$=4$$ square units.

  • Question 4
    1 / -0
    If there are five consecutive integer in a series and the first integer is $$1$$, what is the value of the last consecutive integer?
    Solution
    Let the five consecutive integers be $$x, x + 1, x + 2, x + 3$$ and $$x + 4$$.
    Therefore, the value of first integer is $$1$$ i.e., $$x = 1$$
    So, the last integer is $$x + 4$$
    The value of last integer $$= 5$$.
  • Question 5
    1 / -0
    In a _______ each term is found by multiplying the previous term by a constant.
    Solution
    In a geometric sequence each term is found by multiplying the previous term by a constant.
  • Question 6
    1 / -0
    A sequence of numbers in which each term is related to its predecessor by same law is called
    Solution

    A sequence of numbers in which each term is related to its predecessor by same law is called progression
    Example: 1, 2, 3, 4.... is an example of sequence or progression.
    Since the given sequence follows a same rule or law through out the sequence and there is a relation between each term and it's previous one.
  • Question 7
    1 / -0
    $$5, 25, 125, .....$$ is an example of 
    Solution
    In geometric progression, the ratio of consecutive terms should be equal.
    Here $$5, 25, 125, ...$$ is an example of geometric progression as their common ratio is $$5$$.
  • Question 8
    1 / -0
    A ______ is the sum of the numbers in a geometric progression.
    Solution
    A geometric series is the sum of the numbers in a geometric progression.
  • Question 9
    1 / -0
    Identify the geometric series.
    Solution
    Geometric series is of the following form:
    $$a+ar+ar^2+ar^3 +ar^4+..........+ar^n$$
    Series $$2+12+72+432+......$$ follows the same with $$a=2$$ and $$r=6$$.
    Hence, option B is correct.
  • Question 10
    1 / -0
    A _________ is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a unchanging number called the common ratio.
    Solution
    A geometric progression is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a with a unchanging number called the common ratio.
    Example: $$2, 6, 18, 54, 108....$$
    This geometric sequence has a common ratio $$3$$. 
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