Question

The sum of all even natural numbers less than hundred is

Answer 1

Answer:

All Natural Even Numbers less than 100 start from 2 and continue till 98.

We can observe that this series forms an Arithmetic Progression with a common difference of 2.

Hence the sum can be calculated by using the Sum of an AP formula.

• a = 2
• d = 2
• l = 98
• n = ?

According to the 'n'th term formula,

⇒ l = a + ( n - 1 ) d  

where, 'l' is the last term of the AP, 'a' is the first term, 'd' is the common difference and 'n' is the number of terms in an AP.

Substituting in the formula we get,

⇒ 98 = 2 + ( n - 1 ) 2

⇒ 98 - 2 = ( n - 1 ) 2

⇒ 96 = 2 ( n - 1 )

⇒ 96/2 = ( n - 1 )

⇒ 48 = n - 1

⇒ n = 48 + 1 = 49

Therefore number of terms is 49.

Sum of an AP = n/2 [ a + l ]

⇒ Sum = 49/2 [ 2 + 98 ]

⇒ Sum = 49/2 ( 100 )

⇒ Sum = 2450

Therefore the sum of all even natural numbers less than 100 is 2450.

Answer 2

$\huge{\underline{\underline{\purple{\sf{Answer :}}}}}$

★ Given :

A.P : 2, 4, 6, 8, 10 .............. 98

First tern (a) = 2

Common Difference (d) = 2

Last term (An or L) = 98

_________________________

★ To Find :

Sum of all even terms less than 100.

_________________________

★ Solution :

We know that,

$\Large{\boxed{\boxed{\green{\sf{A_{n} = a + (n - 1)d}}}}}$

98 = 2 + (n - 1)2

98 = 2 + 2n - 2

98 = 2n

98/2 = n

49 = n

$\large{\boxed{\blue{\sf{Number \: of \: terms \: (n) = 49}}}}$

$\rule{200}{2}$

Now,

$\Large{\boxed{\boxed{\pink{\sf{S_{n} = \frac{n}{2}(a + L)}}}}}$

(Putting Values)

$\sf{s_{n} = \frac{49}{2}(2 + 98) } \\ \\ \sf{s_{n} = \frac{49}{ \cancel{2}}( \cancel{100}) } \\ \\ \sf{s_{n} = 49 \times 50 } \\ \\ \sf{s_{n} = 2450}$

$\large{\boxed{\red{\sf{S_{n} = 2450}}}}$

$\rule{200}{2}$

♦ Additional information :

$\boxed{\begin{minipage}{7 cm} Formula's related to A.P \\ \\ A_{n} = a + (n - 1)d \\ \\ S_{n} = \frac{n}{2}2a + (n - 1)d \\ \\ S_{n} = \frac{n}{2}(a + L) \end{minipage}}$

$\rule{200}{2}$

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