Question

Two AP's have the same common difference. The
difference between their 100thterms is 100, what is
the difference between their 1000th terms​

Answer 1

$\underline{\underline{\huge{\gray{\tt{\textbf Answer :-}}}}}$

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$\sf\large\bold{\orange{\underline{\blue{ Given :-}}}}$

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• Two AP's have the same common difference

• The difference between their 100th terms is 100

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$\sf\large\bold{\orange{\underline{\blue{ To \: Find :-}}}}$

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• The difference between their 1000th terms

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$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

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We know that ,

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$\underline{ \underline{\bold{\texttt{For nth term :}}}}$

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➠ $\bf a_n = a + (n - 1)d$ ⚊⚊⚊⚊ ⓵

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Where ,

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• $\sf a_n$ = nth term

• a = First term

• n = Number of terms

• d = Common difference

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Case I [ 1st AP ]

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Let a be the first term in 1st AP and common difference be d

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$\underline{ \underline{\bold{\texttt{For 100th term :}}}}$

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• $\sf a_n = a_{100}$

• a = a

• n = 100

• d = d

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⟮ Putting the above values in ⓵ ⟯

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: ➜ $\sf a_n = a + (n - 1)d$

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: ➜ $\sf a_{100} = a + (100 - 1)d$

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: ➜ $\sf a_{100} = a + 99d$ ⚊⚊⚊⚊ ⓶

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$\underline{ \underline{\bold{\texttt{For 1000th term :}}}}$

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• $\sf a_n = a_{1000}$

• a = a

• n = 1000

• d = d

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⟮ Putting the above values in ⓵ ⟯

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: ➜ $\sf a_n = a + (n - 1)d$

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: ➜ $\sf a_{1000} = a + (1000 - 1)d$

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: ➜ $\sf a_{1000} = a + 999d$ ⚊⚊⚊⚊ ⓷

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Case II [ 2nd AP ]

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Let a' be the first term in 2nd AP and common difference be d as that of 1st AP

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$\underline{ \underline{\bold{\texttt{For 100th term :}}}}$

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• $\sf a_n = a'_{100}$

• a = a'

• n = 100

• d = d

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⟮ Putting the above values in ⓵ ⟯

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: ➜ $\sf a_n = a + (n - 1)d$

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: ➜ $\sf a'_{100} = a' + (100 - 1)d$

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: ➜ $\sf a'_{100} = a' + 99d$ ⚊⚊⚊⚊ ⓸

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$\underline{ \underline{\bold{\texttt{For 1000th term :}}}}$

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• $\sf a_n = a'_{1000}$

• a = a'

• n = 1000

• d = d

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⟮ Putting the above values in ⓵ ⟯

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: ➜ $\sf a_n = a + (n - 1)d$

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: ➜ $\sf a'_{1000} = a' + (1000 - 1)d$

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: ➜ $\sf a'_{1000} = a' + 999d$ ⚊⚊⚊⚊ ⓹

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Given that , the difference between their 100th terms is 100

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Thus ,

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⟮ Equation ⓶ - ⓸ = 100 ⟯

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: ➜ $\sf a_{100} - a'_{100} = a + 99d - (a' + 99d)$

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: ➜ $\sf a + 99d - a' - 99d = 100$

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: ➜ $\sf a - a' = 100$ ⚊⚊⚊⚊ ⓺

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We need to calculate the difference between their 1000th terms

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⟮ i.e Equation ⓷ - ⓹ ⟯

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: ➜ $\sf a_{1000} - a'_{1000} = a + 999d - (a' + 999d)$

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: ➜ $\sf a_{1000} - a'_{1000} = a + 999d - a' - 999d$

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: ➜ $\sf a_{1000} - a'_{1000} = a - a'$ ⚊⚊⚊⚊ ⓻

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⟮ Putting a - a' = 100 from ⓺ to ⓻ ⟯

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: ➜ $\sf a_{1000} - a'_{1000} = a - a'$

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: : ➨ $\sf a_{1000} - a'_{1000} = 100$

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• Hence the difference between their 1000th terms is 100

Answer 2

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$\sf \blue{Question:}$

Two AP's have the same common difference. The

difference between their 100thterms is 100, What is the difference between their 1000th terms.

$\sf \orange{Answer:}$

• Difference between 1000th terms = 100

$\sf \green{Given:}$

• The both AP's have the same common difference.
• Difference between their 100th terms is 100.

$\sf \purple{To \: calculate:}$

• Difference between their 1000th terms

$\sf \orange{Explanation:}$

$\boxed{ \sf Formula = a_n = a + (n - 1)d}$

• n = 100

$\sf \green{According \: to \: the \: question:}$

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Case - I

Let us find 100th term of 1st AP:

• a = 1st AP

$\sf a_{100} = a + (100 - 1)d$

$\sf a_{100} = a + 99d$

Thus , 100th term of 1st AP = a + 99d

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Case - II

100th term of 2nd AP:

• b = 2nd AP

$\boxed{ \sf Formula = b_n = b + (n - 1)d}$

$\sf b_{100} = b + (100 - 1)d$

$\sf b_{100} = b + 99d$

Thus , 100th term of 1st AP = b + 99d

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As Question is given as:

The difference between their 100th terms is 100.

$\sf a_{100} - b_{100} = 100$

(a + 99d) - (b + 99d) = 100

a - b + 99d - 99d = 100

a - b = 100 ... (eq - 1)

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At last,

Let us find the difference between their 1000th terms:

$\sf a_{1000} - b_{1000} = 100$

$\rightarrow \sf [ a - (1000 - 1)d ] - [ b - (1000 - 1)d ]$

$\rightarrow \sf a + 999d - b - 999d$

$\rightarrow \sf a - b + 999d - 999d$

$\rightarrow \sf a - b + \cancel{999d} - \cancel{999d}$

a - b = 100

• Putting a - b = 100 from eq - 1.

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$\sf \blue{Hence:}$

Difference between 1000th terms = 100

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