[1 Mark Question Answer]

Question 11 Mark

Find the sum of the following A.P.s : 2, 7, 12, … to 10 terms

Answer

$2,7,12, \ldots$ to 10 terms
Here $a=2, d=7-2=5$ and $n=10$
$
\begin{aligned}
& S_{10}=\frac{n}{2}[2 a+(n-1) d] \\
& =\frac{10}{2}[2 \times 2+(10-1) \times 5] \\
& =5(4+45) \\
& =5 \times 49 \\
& =245 .
\end{aligned}
$

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Question 21 Mark

Which term of the A.P. 7, 13, 19, … is 205?

Answer

$7,13,19, \ldots$ is 205
Let nth term is 205
$
\begin{aligned}
& \text { Here, } a=7, d=13-7=6 \\
& 205=a+(n-1) d \\
& \Rightarrow 205=7+n(n-1) \times 6 \\
& \Rightarrow 205=7+6 n-6 \\
& \Rightarrow 6 n=205-7+6=204 \\
& n=\frac{204}{6}=34 \\
& \therefore 205 \text { is 34th term. }
\end{aligned}
$

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Question 31 Mark

Which term of the A.P. 3, 8, 13, 18, … is 78?

Answer

$
3,8,13,18, \ldots \text { is } 78
$
Let 78 is nth term
Here, $a=3, d=8-3=5$
$
\begin{aligned}
& \therefore 78=a+(n-1) d \\
& \Rightarrow 78=3+(n-1) 5 \\
& \Rightarrow 78=3+5 n-5 \\
& \Rightarrow 78+5-3=5 n \\
& \Rightarrow 5 n=80 \\
& \Rightarrow n=\frac{80}{5}=16
\end{aligned}
$
$\therefore 78$ is 16 th term.

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Question 41 Mark

If the first term of an A.P. is – 18 and its 10th term is zero, then find its common difference.

Answer

$
\begin{aligned}
& \text { First term }(a)=-18 \\
& T_{10}=0 \\
& a+(n-1) d=T_n \\
& -18+(10-1) d=0 \\
& -18+9 d=0 \\
& \Rightarrow 9 d=18 \\
& \Rightarrow d=\frac{18}{9}=2
\end{aligned}
$
$\therefore$ Common difference $=2$.

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Question 51 Mark

If the common difference of an A.P. is $– 3$ and the 18th term is $– 5,$ then find its first term.

Answer

Common difference $(d) = -3$
$T_{18} = -5$
$a + (n – 1 )d = Tn$
$a + (18 – 1) (-3) = -5$
$\Rightarrow a + 17(–3) = –5$
$\Rightarrow a – 51 = –5$
$\Rightarrow a = –5 + 51 = 46$
$\therefore$ First term $= 46.$

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Question 61 Mark

Find the 8th term of the A.P. whose first term is $7$ and common difference is $3.$

Answer

First term $(a) = 7$
and common difference $(d) = 3$
A.P. $= 7, 10, 13, 16, 19, …$
$T_8 = a + (n – 1)d$
$= 7 + (8 – 1) x 3$
$= 7 + 7 \times 3$
$= 7 + 21$
$= 28$

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Question 71 Mark

If the numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n.

Answer

$\begin{aligned} & n-2,4 n-1 \text { and } 5 n+2 \text { are in A.P. } \\ & \therefore 2(4 n-1)=n-2+5 n+2 \\ & 8 n-2=6 n \\ & \Rightarrow 8 n-6 n=2 \\ & \Rightarrow 2 n=1 \\ & \Rightarrow n \frac{2}{1}=1 \\ & \therefore n=1 .\end{aligned}$

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Question 81 Mark

Find the indicated terms in each of following A.P.s: $1, 6, 11, 16, …; a_{20}$

Answer

$1, 6, 11, 16, …$
Here, $a = 1, d = 6 – 1 – 5$
$a_{20} = a + (n – 1 )d$
$= 1 + (20 – 1) \times 5$
$= 1 + 19 \times 5$
$= 1 + 95$
$= 96$

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Question 91 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : 1, 3, 9, 27,...

Answer

1, 3, 9, 27,...
Here, first term (a) = 1
d = 3 – 1 = 2
9 – 3 = 6
27 – 9 = 18
∵ Common difference is not same.
∴ It is not an A.P.

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Question 101 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : $1^2, 3^2, 5^2, 7^2,...$

Answer

$1^2, 3^2, 5^2, 7^2,...$
$= 1, 9, 25, 49,...$
Here, first term $(a) = 12 = 1$
$d = 9 – 1 = 8$
$25 – 9 = 16$
$49 – 25 = 24$
∵ Common difference is not same.
∴ It is not an A.P.

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Question 111 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : – 10, – 6, – 2, 2,….

Answer

– 10, – 6, – 2, 2,….
Here first term (a) = –10
d = –6 – (–10) = –6 + 10 = 4
–2 – (–6) = –2 + 6 = 4
2 – (–2) = 2 + 2 = 4
∴ Common difference is same.
∴ It is an A.P.
and next three terms are 6, 10, 14, 20.

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Question 121 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P ${ }_t$, find the common difference d and write the next three terms : $2, \frac{5}{2}, 3, \frac{7}{2}, \ldots$

Answer

$
2, \frac{5}{2}, 3, \frac{7}{2}, \ldots
$
Here $a=2$
$
\begin{aligned}
& d=\frac{5}{2}-2=\frac{1}{2} \\
& 3-\frac{5}{2}=\frac{1}{2} \\
& \frac{7}{2}-3=\frac{1}{2}
\end{aligned}
$
$\because$ common difference is same.
$\therefore$ It is an A.P.
and next three terms are $4, \frac{9}{2}, 5$.

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Question 131 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : 2, 4, 8, 16,….

Answer

2, 4, 8, 16,….
Here, a = 2
d = 4 – 2 = 2,
8 – 4 = 4,
16 – 8 = 8
∴ common difference is not same
∴ It is not an A.P.

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Question 141 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : – 2, 2, – 2, 2,…..

Answer

– 2, 2, – 2, 2,…
Here, a = –2
d = 2 – (–2)
= 2 + 2 = 4
–2 – 2 = –4
2 – (–2) = 4
∵ common difference is not same.
∴ It is not an A.P.

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Question 151 Mark

Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : 4, 10, 16, 22,…

Answer

4, 10, 16, 22,…
Here a = 4, d = 10 – 4 = 6, 16 – 10 = 6, 22 – 16 = 6
∵ common difference is same
∵ It is in A.P
and next three terms are 28, 34, 40.

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Question 161 Mark

Write first four terms of the A.P. when the first term a and the common difference $d$ are given as follows : $a =\frac{1}{2}, d =-\frac{1}{6}$

Answer

$\begin{aligned} & a =\frac{1}{2}, d =-\frac{1}{6} \\ & \text { A.P. is } \frac{1}{2},\left(\frac{1}{2}, \frac{1}{6}\right)=\frac{2}{6} \\ & =\frac{2}{6}-\frac{1}{6}=\frac{1}{6}, \ldots \\ & \text { A.P. }=\frac{1}{2}, \frac{2}{6}, \frac{1}{6}, 0, \ldots \\ & =\frac{1}{2}, \frac{1}{3}, \frac{1}{6}, 0, \ldots\end{aligned}$

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Question 171 Mark

Write first four terms of the A.P., when the first term a and the common difference d are given as follows : a = 4, d = – 3

Answer

a = 4, d = – 3
∴ A.P. = 4, 1, –2, –5,...

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Question 181 Mark

Write first four terms of the A.P., when the first term a and the common difference d are given as follows : a = – 2, d = 0

Answer

a = –2, d = 0
∴ A.P. = –2, –2, –2, –2, ….

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Question 191 Mark

Write first four terms of the A.P., when the first term a and the common difference d are given as follows : a = 10, d = 10

Answer

a = 10, d = 10
∴ A.P. = 10, 20, 30, 40, …

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Question 201 Mark

For the following A.P.s, write the first term a and the common difference d: – 3.2, – 3, – 2.8, – 2.6, …

Answer

– 3.2, – 3, – 2.8, – 2.6, …
Here first term (a) = –3.2
and common difference (d)
= –3 – (–3.2) = –3 + 3.2 = 0.2
= (d) = 0.2.

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Question 211 Mark

For the following A.P.s, write the first term a and the common difference $d : \frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots$

Answer

$
\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots
$
Here first common term (a) $=\frac{1}{3}$
and common difference $( d )=$
$
\begin{aligned}
& \frac{5}{3}-\frac{1}{3}=\frac{4}{3}, \frac{9}{3}-\frac{5}{3}=\frac{4}{3}, \ldots \\
& =\frac{4}{3} .
\end{aligned}
$

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Question 221 Mark

For the following A.P.s, write the first term a and the common difference d: 3, 1, – 1, – 3, …

Answer

3, 1, -1, -3, …
Here first term (a) = 3
and the common difference (d)
= 1 – 3 = -2,
– 1 – 1 = -2,…
= -2

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