Name: Sum of First n Terms of an Arithmetic Progression
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: 13+13+13+13…….How do we find the sum of 50 terms of this series? Learn to find the sum of the first ‘n’ terms of an Arithmetic Progression with a real-life e...

Question

According to the above pattern, what is $2 + 4 + 6 + \dots + 30 ?$

Accepted Answer

We have to find the value of $2 + 4 + 6 + \dots + 30$

So, we have first term $= 2$ , the second term $= 4$ , the third term $= 6$ and the last term $= 30$

Therefore, $a_{1} = 2, a_{2} = 4, a_{3} = 6$

Now, the differences between two consecutive terms are,

$d_{1} = a_{2} - a_{1} = 4 - 2 = 2$

$d_{2} = a_{3} - a_{2} = 6 - 4 = 2$

Since, $d_{1} = d_{2}$ i.e., the difference between two consecutive terms is constant i.e., $d = 2$

Therefore, $2, 4, 6, . . . . . . . ., 30$ forms an A.P.

We have to find the number of terms an A.P: $2, 4, 6, . . ., 30$

We know that, the $n^{t h}$ term of an A.P is, $a_{n} = a + (n - 1) d$ , where $a$ is the first term, $d$ is the common difference and $a_{n}$ is the last term.

We have $a = 2, d = 2 and a_{n} = 30$

Now,

$30 = 2 + (n - 1) 2$

$\Rightarrow 30 = 2 + 2 n - 2$

$\Rightarrow 30 = 2 n$

$\Rightarrow n = 15$

We know that, the sum of first $n$ terms of an A.P is $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$

Now, we have to find the sum of first $15$ terms of an A.P.

Therefore, $S_{15} = \frac{15}{2} (2 \times 2 + (15 - 1) 2)$

$\Rightarrow S_{15} = \frac{15}{2} (4 + 14 \times 2)$

$\Rightarrow S_{15} = \frac{15}{2} (32)$

$\Rightarrow S_{15} = 15 \times 16 = 240$

Therefore, $2 + 4 + 6 + \dots + 30$ $= 15 \times 16 = 240$

According to the above pattern, what is $2 + 4 + 6 + \dots + 30 ?$

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

According to the above pattern, what is 2+4+6+…+30?

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

According to the above pattern, what is $2 + 4 + 6 + \dots + 30 ?$