Question

calculate the sum of first 20 terms in the arithmetic sequence 2n+3​

Answer 1

$\underline{\pmb{\sf{\; Understanding\; the\; question\;:}}}$

Here, the concept of Arithmetic progressions has been used. We're given with the nth term of an AP and we've been asked to find the sum of first 20 terms. So, in order to solve this sum, we firstly need to to calculate the first term and common difference of the arithmetic progression. So, firstly we are going to assume the value of n and we'll find the first two terms of the arithmetic progression by assuming value of n (the value of n we are going to take should be consecutive) and we know the common difference is the difference between it's second and first term. After calculating the first term and common difference, we will find the sum of first 20 numbers by using it's formula. Let's examine the formula for sum of n terms.

$\underline{\pmb{\sf{\; Formula\; used\;:}}}$

» Here, we're asked to calculate the sum of n terms. The sum of n terms is calculated by finding the product of n/2 and addition of twice of the first term (n-1) multiplied by common difference. So, writing this in the form of formula—

$\qquad\star~{\underline{\boxed{\pmb{\frak{{S}_{n}=\dfrac{n}{2}\bigg(2a+(n-1)d\bigg)}}}}}$

where:

• $\bf{{S}_{n}}$ is the sum of n terms.
• n is the term.
• a is the first term.
• d is the common difference.

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$\underline{\pmb{\sf{\; Solution\;:}}}$

It is stated in the question that $\bf{{a}_{n}}$ is 2n+3. We're asked to calculate the sum of first 20 terms in AP. So, firstly let's calculate the first term and common difference.

$\star$ Let us assume the value of n as 1. So,

$:\implies\quad\sf{{a}_{1}=2n+3}$

$\\:\implies\quad\sf{{a}_{1}=2(1)+3}$

$\\:\implies\quad\pmb{\sf{{a}_{1}=5}}$

$\star$ Let's assume that the value of n is 2. So,

$:\implies\quad\sf{{a}_{2}=2n+3}$

$\\:\implies\quad\sf{{a}_{2}=2(2)+3}$

$\\:\implies\quad\pmb{\sf{{a}_{2}=7}}$

So, we've the following data:

• First term, a = 5
• Common difference, d = a2-a1 = 7-5 = 2

$\star$ Now, let's use the formula in order to calculate the sum of first 20 terms in the arithmetic sequence.

$\dashrightarrow\quad\sf{{S}_{20}=\dfrac{n}{2}\bigg(2a+(n-1)d\bigg)}$

$\\\dashrightarrow\quad\sf{{S}_{20}=\dfrac{20}{2}\bigg(2\times 5+(20-1)2\bigg)}$

$\\\dashrightarrow\quad\sf{{S}_{20}=10\bigg(10+(19)2\bigg)}$

$\\\dashrightarrow\quad\sf{{S}_{20}=10\bigg(10+38\bigg)}$

$\\\dashrightarrow\quad\sf{{S}_{20}=10\times 48}$

$\\\dashrightarrow\quad\underline{\boxed{\pmb{\frak{{S}_{20}=480}}}}$

$\\\boxed{\rm{Hence,\; sum\; of\; first\; 20\; numbers\; is\;}\pmb{480}}$

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