Question

2+3+5+6+8+9+......+(3n-1)+(3n)


answer=n (3n+2)

Answer 1

in the Series 2+3+5+6+8+9....

there are two series
1st) 2 + 5 + 8 ......3n-1

2nd ) 3 + 6 + 9.....3n


Sum of 1st series = [n(2 + 3n-1) ]/ 2

Sum of 2nd series = [n(3 + 3n)]/2

Total sum = n(3n+2)

Answer 2

Answer:The $(a_{n} + 2)th$ term of the given series is $\frac{7}{2}$.

Step-by-step explanation:

Given:We have given series 2+3+5+6+7+8+9+......+(3n-1)+(3n)

To find:We have to find the the sum of the series.

Explanation:

Step 1: First term in the series $a=2$

            Common difference $d = 1$

            Last term in the series $a_{n} = 3n$

Step 2:As  $a_{n} = 3n$ and we know that $a_{n} = a+(n-1)d$. So, we have,

⇒                               $a_{n} = 3n$

⇒               $a+(n-1)d = 3n$

⇒               $2+(n-1)1 = 3n$

⇒                    $2+n-1 = 3n$

⇒                          $n+1 = 3n$

⇒                                 $1 = 3n-n$

⇒                                 $1 = 2n$

∴                                  $n = \frac{1}{2}$

Step 3:The $(a_{n} + 2)th$ term of the given series can be given by,

⇒                    $a_{n} +2 = 3n+2$

⇒                               $= 3(\frac{1}{2}) + 2$

⇒                               $= \frac{3}{2}+ 2$

⇒                               $= \frac{3+4}{2}$

⇒                               $= \frac{7}{2}$

∴The $(a_{n} + 2)th$ term of the given series is $\frac{7}{2}$.

Concept:Arithmetic Progression (AP) is a series of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also called Arithmetic Sequence. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an Arithmetic Progression, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.

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