Arithmetic Progressions Test

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Arithmetic Progressions Test - 58

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Weekly Quiz Competition

Question 1
1 / -0

The formula to find $$n^{th}$$ term of an A.P is,

A
$$a + (n - 1)d$$

B
$$ar^{n - 1}$$

C
$$\dfrac{1}{a + (n - 1)d}$$

D
$$a - (n + 1)d$$

Solution

nth term of the A.P. is calculated by the formula,
$${ a }_{ n }=a+\left( n-1 \right) d$$

where, $$a$$ = first term of A.P.
$$n$$ = total number of terms in A.P.
$$d$$ = common difference between two consecutive terms of A.P.
Question 2
1 / -0

In an A.P, if $$a_{4} = 8 \,\& \,a = 2,$$ then its common difference is

A
$$6$$

B
$$4$$

C
$$2$$

D
$$10$$

Solution

nth term of the A.P. is given by,
$${ a }_{ n }=a+\left( n-1 \right) d$$

$$\therefore { a }_{ 4 }=a+\left( 4-1 \right) d$$

$$\therefore 8=2+3d$$ (Given)

$$\therefore 3d=6$$

$$\therefore d=2$$

Thus, common difference is 2.
Question 3
1 / -0

If sum of $$n$$ term of $$A.P.$$ is $$S_{n}$$ and $$S_{2n}=3S_{n}$$, then $$S_{3n}: S_{n}$$ will be:

A
$$10$$

B
$$11$$

C
$$6$$

D
$$4$$

Solution

$$S_{2n}=3 S_{n}$$
$$\dfrac{2n}{2}[2a(2n-1)d]=\dfrac{3n}{2}[2a+(n-1)d]$$
$$\Rightarrow 4a+4nd-2d=6a+3nd-3d$$
$$\Rightarrow nd+d=2a$$
Now, $$S_{3n}:S_{n}$$
$$\dfrac{S_{3n}}{S_{n}}=\dfrac{\dfrac{3n}{2}[2a+(3n-1)d]}{\dfrac{n}{2}[2a+(n-1)d]}$$
$$\dfrac{S_{3n}}{S_{n}}=\dfrac{3[nd+d+3nd-d]}{[nd+d+nd-d]}$$
$$[\because 2a=nd+d]$$
$$=\dfrac{3\times 4nd}{2\ nd}$$
$$=\dfrac{12}{2}$$
$$=6$$
Hence, option $$(C)$$ is correct.
Question 4
1 / -0

If sum of $$n$$ terms of $$A.P$$ is $$3n^2+5n$$, then its which term is $$164$$:

A
$$12^{th}$$

B
$$15^{th}$$

C
$$27^{th}$$

D
$$20^{th}$$

Solution

Given: $$S_{n}=3n^{2}+5n$$
$$S_{1}=3(1)^{2}+5(1)=8$$
$$S_{2}=3(2)^{2}+5(2)=22$$
$$S_{3}=3(3)^{2}+5(3)=42$$
$$S_{4}=3(4)^{2}+5(4)=68$$
$$\therefore a_{1}=S_{1}=8$$
$$a_{2}=S_{2}-S_{1}\Rightarrow 22-8\Rightarrow 14$$
$$a_{3}=S_{2}-S_{1}\Rightarrow 42-22\Rightarrow 20$$
$$a_{4}=S_{2}-S_{1}\Rightarrow 68-42\Rightarrow 26$$
Thus $$A.P.$$ will be $$8,14,20,26,......164$$
$$a=8$$,
$$d=14-8=6$$ and $$a_{n}=164$$
$$\therefore 164=a+(n-1)d$$
$$164=8+(n-1)$$
$$(n-1)=156/6=26$$
$$\therefore n=26+1=27$$
Hence, option $$(C)$$ is correct.
Question 5
1 / -0

A manufacture company made $$15000$$ mobile phones in its $$17^{th}$$ year. The production has been increasing by $$500$$ every year since the first year.
How many mobile phones did the company make in second year?

A
$$7000$$

B
$$7500$$

C
$$8000$$

D
$$8500$$

Solution
Question 6
1 / -0

In the A.P. $$2, -2, -6, -10........$$ common difference (d) is:

A
$$-4$$

B
$$2$$

C
$$-2$$

D
$$4$$

Solution

Common difference $$d= -2-2$$ $$= -6-(-2) $$ { $$t_{n+1}-t_n=d$$ }
$$= -4$$
Question 7
1 / -0

If we divide $$18$$ into three parts that are in $$AP$$ and whose product is $$120,$$ then formed $$AP$$ is given by

A
$$2, 7, 9$$

B
$$2, 5, 11$$

C
$$2, 6, 9$$

D
$$2, 4, 10$$

Solution
Question 8
1 / -0

Dishan started reading a novel which had $$240$$ pages. At the end of the first day, she was on page number $$45$$. She slowed after that. She read $$39$$ pages every day after the first day. How long did it take Dishan to complete that novel?

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$
Question 9
1 / -0

A roll of thread $$264$$ $$cm$$ long is cut into four parts to make four circles. The radii of the circles increases by $$1$$ $$cm$$ consecutively. Find the radius of smallest circle?

A
$$9$$

B
$$11$$

C
$$12$$

D
$$13$$
Question 10
1 / -0

Find an $$AP$$ whose general term is given by $$3n+1$$.

A
$$4, 7, 10, 13, .....$$

B
$$3, 6, 9, 12, .....$$

C
$$3, 7, 12, 15, .....$$

D
$$2, 5, 9, 13, .....$$

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Re-Attempt Weekly Quiz Competition

Arithmetic Progressions Test - 58

Result

Arithmetic Progressions Test - 58

Score

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Rank

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SHARING IS CARING

Question 1

$$a + (n - 1)d$$

$$ar^{n - 1}$$

$$\dfrac{1}{a + (n - 1)d}$$

$$a - (n + 1)d$$

Solution

Question 2

$$6$$

$$4$$

$$2$$

$$10$$

Solution

Question 3

$$10$$

$$11$$

$$6$$

$$4$$

Solution

Question 4

$$12^{th}$$

$$15^{th}$$

$$27^{th}$$

$$20^{th}$$

Solution

Question 5

$$7000$$

$$7500$$

$$8000$$

$$8500$$

Solution

Question 6

$$-4$$

$$2$$

$$-2$$

$$4$$

Solution

Question 7

$$2, 7, 9$$

$$2, 5, 11$$

$$2, 6, 9$$

$$2, 4, 10$$

Solution

Question 8

$$5$$

$$6$$

$$7$$

$$8$$

Question 9

$$9$$

$$11$$

$$12$$

$$13$$

Question 10

$$4, 7, 10, 13, .....$$

$$3, 6, 9, 12, .....$$

$$3, 7, 12, 15, .....$$

$$2, 5, 9, 13, .....$$

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