Question

The nth term of an AP is 5n - 3. Find out the sum of its first 25 terms.​

Answer 1

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• The nth term of an AP is 5n - 3

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The sum of first 25 term of the Arithmetic Progression.

$\bf{\underline{\underline\green{Solution:-}}}$

Given:-

nth term = 5n - 3

To get the First term substitute n = 1

➝ a = 5(1) - 3

➝ 5 - 3

➝ 2

Second term :-

Substitute n = 2

➝ $\rm a_{2} = 5(2) - 3$

➝ 10 - 3

➝ 7

Common difference

$\rm = a_{2} - a_{1}$

= 7 - 2

= 5

Now we have

• a = 2 and d = 5

As we know that

Sum of n terms is given by the formula

$\boxed{ \rm \leadsto S_{n} = \frac{n}{2} \bigg \{2a + (n - 1)d \bigg \}}$

Sum of first 25 terms:-

Here:-

• n = 25

• a = 2

• d = 5

$\rm \leadsto S_{25} = \dfrac{25}{2} \bigg \{2 \times 2+ (25 - 1)5 \bigg \}$

$\rm = \dfrac{25}{2} \bigg \{4+ (24 \times 5 )\bigg \}$

$\rm = \dfrac{25}{2} \bigg \{4+ 120\bigg \}$

$\rm = \dfrac{25}{2} \times 124$

$\rm = 1550$

$\underline{ \boxed{\rm \therefore Sum \: of \: 25 \: terms = 1550}}$

Answer 2

Given :

• n th term of AP = 5 n - 3

To find :

• Sum of first 25 terms []

Formula to be used :

$\bigstar \: \bf{ \boxed{\mathtt{s = \frac{n}{2} [2a + (n - 1)d ]}}}$

here ,

• a = first term
• d = difference between the two terms in an AP
• n = n th term

Concept :

• Let a be the first term
• From the given equation first we need to find the first and the second terms of AP in order to find d
• Then substitute the values of a and d in the above equation in order to find out the sum .

Solution :

n the term of an AP = 5 n - 3

____________________________________

First term :

Substituting n = 1 we get ,

>> a = 5 n - 3

>> a = 5 - 3

>> a = 2

_____________________________________

Second term :

Substituting n = 2 we get ,

>> a 1 = 5 n - 3

>> a 1 = 5×2 - 3

>> a 1 = 7

_____________________________________

Difference between the second and the first term

>> d = a 1 - a

>> d = 7 - 2

>> d = 5

______________________________________

$\bigstar \: \bf{ \boxed{\mathtt{s = \frac{n}{2} [2a + (n - 1)d ]}}}$

here ,

• a = 2
• d = 5
• n = 25

$\rightarrow \: \bf {\mathtt{s = \frac{25}{2} [2 \times 2 + (25- 1)5 ]}}$

$\rightarrow \: \bf {\mathtt{s = 12.5 [4 + 24 \times 5 ]}}$

$\rightarrow \: \bf {\mathtt{s = 12.5 [124 ]}}$

$\rightarrow \: \bf \boxed{ \red {\mathtt{s = 1550}}}$

Sum of the first 25 terms of AP is 1550