Question

How many three digit numbers are divisible by 7 ?
48. Find the sum of terms of the A.P. : 4, 9, 14, ............, 89.

Answer 1

Answer:

$\large{\underline{\boxed{\sf (i) \: \: 128}}}$

$\large{\underline{\boxed{\sf (ii) \: \: 837}}}$

Step-by-step explanation:

(i) The 3 - digit numbers divisible by 7 are as follows ;

105, 112, 119, ......, 994.

Clearly, these numbers form an AP with,

• a = 105
• d = (112 - 105) = 7
• last term = 994

Let the total number of terms be n. Then,

Tₙ = 994

⇒ a + (n - 1)d = 994

⇒ 105 + (n - 1) * 7 = 994

⇒ 105 + 7n - 7 = 994

⇒ 7n + 98 = 994

⇒ 7n = 994 - 98

⇒ 7n = 896

⇒ n = $\sf \dfrac{896}{7}$

⇒ n = 128

Hence, there are 128 three-digit numbers divisible by 7.

$\rule{300}{2}$

(ii) Find the sum of terms of the AP : 4, 9, 14,....., 89.

Here,

• a = 4
• d = (9 - 4) = 5
• l = 89

Let the total number of terms be n. Then,

Tₙ = 89

⇒ a + (n - 1)d = 89

⇒ 4 + (n - 1) * 5 = 89

⇒ 4 + 5n - 5 = 89

⇒ 5n - 1 = 89

⇒ 5n = 89 + 1

⇒ 5n = 90

⇒ n = $\sf \dfrac{90}{5}$

⇒ n = 18

Required sum = $\rm \dfrac{n}{2} \:. \: (a + l)$

⇒ Sum = $\rm \dfrac{18}{2} \:. \: ( 4+ 89)$

⇒ Sum = 9 * 93

⇒ Sum = 837

Hence, the required sum is 837.

Answer 2

Answer:

$\huge{\red{\boxed{\boxed{\green{\sf{\therefore (i)n=128}}}}}}$

$\huge{\red{\boxed{\boxed{\green{\sf{\therefore(ii) Sn=837}}}}}}$

Step-by-step explanation:

$\huge{\red{\boxed{\boxed{\green{\underline{\red{\sf{SOLUTION-}}}}}}}}$

1st QUESTION:

$according \: to \: given \: question \: the \: first \\ \: three \: digit \: smallest \: number \: that \\ can \: be \: divied \: by \: 7 \: is \: first \: term. \\ \\ {\green {\boxed{a = 105}}} \\ {\green { \boxed{d =7}}} \\ { \green{\boxed{an = 994}}} \\ \\ an = a + (n - 1)d \\ = ) 994 = 105 + (n - 1) \times 7 \\ = )994 - 105 = (n - 1) \times 7 \\ = ) 889 = (n - 1) \times 7 \\ = ) \frac{889}{7} = n - 1 \\ = )127 = n - 1 \\ = )n = 127 + 1 \\ = )n = 128$

$\huge{\red{\boxed{\boxed{\green{\sf{\therefore n=128}}}}}}$

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2nd QUESTION:

${ \green{a = 4 }} \\ {\green{d = 5}} \\ {\green{an = 89}} \\ {\red{n = }}\\ {\red{ sn = }} \\ \\ an = a +( n - 1)d \\ = )89 = 4 + (n - 1) \times 5 \\ = )89 - 4 =( n - 1) \times 5 \\ = ) \frac{85}{5} = n - 1 \\ = )n = 17 + 1 \\ = )n = 18 \\ \\ sn = \frac{n}{2} (a + an) \\ s18 = \frac{18}{2} (4 + 89) \\ s18 = 9 \times 93 \\ s18 = 837$

$\huge{\red{\boxed{\boxed{\green{\sf{\therefore Sn=837}}}}}}$

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