Question

find the sum of all the number between 100 and 200 which are divisible by 7?

Answer 1

$\underline{\underline{ \large{ \mathfrak{Solution : }}}}$
$\textsf{Numbers between 100 and 200 which is}\\ \sf divisible \: by \: 7 \longrightarrow 105 , 112 , ..... , 196.$
$\textsf{Here , we can observe that this sequence } \\ \textsf{forms an Arithmetic Progression.}$
$\underline{ \textsf{Where :}} \\ \\ \sf \implies First \: term (a) \: = \: 105 \\ \\ \sf \implies Common \: difference (d) \: = \: 112 \: - \: 105 \: \\ \sf \qquad \qquad \qquad \qquad \qquad \qquad \: \: = \: 7 \\ \\ \sf \implies Last \: term(l) \: = \: 196$
$\textsf{Now,} \\ \\ \sf \implies l \: = \: a \: + \: ( n \: - \: 1 )d \\ \\ \sf \implies 196 \: = \: 105 \: + \: ( n \: - \: 1 )7 \\ \\ \sf \implies 196 \: - \: 105 \: = \: ( n \: - \: 1 )7 \\ \\ \sf \implies 91 \: = \: ( n \: - \: 1)7 \\ \\ \sf \implies 91 \: \div \: 7 \: = \: n \: - \: 1 \\ \\ \sf \implies 13 \: = \: n \: - \: 1 \\ \\ \sf \implies n \: = \: 13 \: + \: 1 \\ \\ \sf \: \therefore \: n \: = \: 14$
$\underline{\textsf{Again, }} \\ \\ \sf \implies S_n \: = \: \dfrac{n}{2}(a \: + \: l) \\ \\ \sf \implies S_{14 }\: = \: \dfrac{14}{2}( 105 \: + \: 196 ) \\ \\ \sf \implies S_{14 }\: = \: 7 \: \times \: 301 \\ \\ \sf \: \therefore \: S_{14 }\: = \: 2107$

Answer 2

hey mate !!

here is your answer !!

in the questions given the sum of all the numbers between 100 and 200 which are divisible by 7 such as 105 , 112, 119, -- , --, --, 196 .

the given sequence terms 105 ,112, 119 - - - 196 it's form of an Ap.

here given,

first term a1 = 105

second term a2 = 112,

third term a3 = 119

$a_{n} term = 196$

common difference ( d) = a2 -a1

d = 112 - 105

d = 7

now we find,

$a_{n} = a ( n - 1 ) d$

196 = 105 ( n - 1 ) 7

196 - 105 = ( n - 1 ) 7

91 = 7 ( n - 1 )

91 = 7n - 7

91 + 7 = 7n

98 = 7n

$n \: = \frac{98}{7} = 14$

n = 14

again we using formula ,

$S_{n} = \frac{n}{2} (a + a_{n}) \\ \\ S_{14} = \frac{14}{2} (105 + 196) \\ \\ S_{14} = \frac{14}{2} (301) \\ \\ S_{14} = 7 \times 301 \\ \\ \\ S_{14} = 2107.$

therefore sum of its n terms equal to 2107.

be brainly