Question

the algebraic form of an arithmetic sequence is 2n+3 find the sum of first 20 terms​

Answer 1

Answer :

$\large \bigstar \: \: \: \boxed {s_{20} = 480} \: \: \: \bigstar$

Explanation :

✦ Given :–

$a_n = 2n + 3$

✦ To Find :–

• $s_{20} \: (sum \: of \: first \: 20 \: terms)$

✦ Formulas Applied :–

• $a_n = a + (n - 1) \times d$
• $s_n = \frac{n}{2} [2a + (n - 1) \times d]$
• $d = a_{(n + 1)} - a_n$

✦ Solution :–

We know that,

$a_n = 2n + 3$

→ Put value of n=1,

$\implies a_1 = 2(1) + 3$

$\implies a_1 = 2 + 3$

$\implies a_1 = 5$

$\bigstar \: \boxed {a = 5}$

→ Put value of n=2,

$\implies a_2 = 2(2) + 3$

$\implies a_2 = 4+ 3$

$\implies \boxed{ a_2 = 7}$

Now we know that,

$d = a_{(n + 1)} - a_n$

$\implies d = a_{(1+ 1)} - a_1$

$\implies d = a_2 - a_1$

$\implies d = 7 - 5$

$\bigstar \: \boxed{ d = 2}$

Now, we have :

• a=5
• d=2
• n=20

Applying the formula:

$s_n = \frac{n}{2} [2a + (n - 1) \times d]$

$\implies s_{20} = \frac{20}{2} [2(5) + (20 - 1) \times (2)]$

$\implies s_{20} = 10 \times [10 + (19) \times (2)]$

$\implies s_{20} = 10 \times (10 + 38)$

$\implies s_{20} = 10 \times 48$

$\star \: \boxed {s_{20} = 480} \: \star$

∴ Sum of First 20 Terms is 480.

→ Additional Information :–

There is one more formula:

$s_n = \frac{n}{2} (a + l) \: where \: a \: is \: first \: term \: and \: l \: is \: last \: term.$

It is similar as the formula:

$s_n = \frac{n}{2} [2a + (n - 1) \times d]$

It is applied only when the first and the last term is given.

[✓ Note : It is better to apply this formula as the calculation will be easier from this formula compared to the first formula which is used in this question.]