(A) Given the recursive formula:$a_{1} = 1$$a_{n} = a_{n-1} + 2$ for $n \ge 2$Calculating the terms:$a_{1} = 1$$a_{2} = a_{1} + 2 = 1 + 2 = 3$$a_{3} =

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$1, 3, 5, 7, 9$ and $1+3+5+7+9+\ldots$

(A) Given the recursive formula:$a_{1} = 1$$a_{n} = a_{n-1} + 2$ for $n \ge 2$Calculating the terms:$a_{1} = 1$$a_{2} = a_{1} + 2 = 1 + 2 = 3$$a_{3} = a_{2} + 2 = 3 + 2 = 5$$a_{4} = a_{3} + 2 = 5 + 2 = 7$$a_{5} = a_{4} + 2 = 7 + 2 = 9$The first five terms are $1, 3, 5, 7, 9$.The corresponding series is $1 + 3 + 5 + 7 + 9 + \ldots$

Class 11 Mathematics Sequences and Series Arithmetic progression

Medium

Let the sequence $a_{n}$ be defined as follows:
$a_{1} = 1, a_{n} = a_{n-1} + 2$ for $n \ge 2$
Find the first five terms and write the corresponding series.

A
$1, 3, 5, 7, 9$ and $1+3+5+7+9+\ldots$
B
$1, 2, 3, 4, 5$ and $1+2+3+4+5+\ldots$
C
$1, 4, 7, 10, 13$ and $1+4+7+10+13+\ldots$
D
$2, 4, 6, 8, 10$ and $2+4+6+8+10+\ldots$

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Sequences and Series

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Arithmetic progression

If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

DifficultJEE MAIN 2013

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Let $a_{1}, a_{2}, \ldots, a_{21}$ be an $A.P.$ such that $\sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}} = \frac{4}{9}$. If the sum of this $A.P.$ is $189$,then $a_{6} a_{16}$ is equal to:

DifficultJEE MAIN 2021

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Let the sum of the first three terms of an $A.P.$ be $39$ and the sum of its last four terms be $178.$ If the first term of this $A.P.$ is $10,$ then the median of the $A.P.$ is

DifficultJEE MAIN 2015

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The number of terms common to the two $A$.$P$.'s $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

DifficultJEE MAIN 2020

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If the sum of $n$ terms of an $A.P.$ is $(pn + qn^2),$ where $p$ and $q$ are constants,find the common difference.

Medium

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Let the sequence $a_{n}$ be defined as follows:$a_{1} = 1, a_{n} = a_{n-1} + 2$ for $n \ge 2$Find the first five terms and write the corresponding series.

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If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

Let $a_{1}, a_{2}, \ldots, a_{21}$ be an $A.P.$ such that $\sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}} = \frac{4}{9}$. If the sum of this $A.P.$ is $189$,then $a_{6} a_{16}$ is equal to:

Let the sum of the first three terms of an $A.P.$ be $39$ and the sum of its last four terms be $178.$ If the first term of this $A.P.$ is $10,$ then the median of the $A.P.$ is

The number of terms common to the two $A$.$P$.'s $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

If the sum of $n$ terms of an $A.P.$ is $(pn + qn^2),$ where $p$ and $q$ are constants,find the common difference.

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Let the sequence $a_{n}$ be defined as follows:
$a_{1} = 1, a_{n} = a_{n-1} + 2$ for $n \ge 2$
Find the first five terms and write the corresponding series.