Question

34.
The first term of two A.P.s are equal and the ratios of their common
differences is 1 : 2. If the 7th term of first A.P. and 21th term of
second A.P. are 23 and 125 respectively. Find two A.P.S.​

Answer 1

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☞ Ap 1 = 5,8,11,14

☞ AP 2 = 5,11,17,23

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✭ The first term of 2 APs are equal

✭ The Ratio of their common difference is 1:2

✭ The 7th term of an AP and 21st term of another AP are 25 and 125

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➢ The two APs?

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➝ Let the First Term of Both AP be a.

➝ Common Difference of First AP be d and, of Second AP be 2d

❍ $\underline{\textsf{7th Term of First AP }}$

$\leadsto\sf T_n = a + (n - 1)d\\\\\leadsto\sf T_7 = a + (7 - 1)d \\\\\leadsto\sf 23 = a + 6d\qquad -eq(1)$

❍ $\underline{\textsf{21st Term of Second AP }}$

$\longrightarrow\sf T_n = a + (n - 1)d\\\\\longrightarrow\sf T_{21} = a + (21 - 1)2d \\\\\longrightarrow\sf 125 = a + 20 \times 2d\\\\\longrightarrow\sf 125 = a + 40d\qquad -eq(2)$

❍ $\underline{\textsf{Subtracting eq(1) from eq(2) }}$

$\hookrightarrow\sf a + 40d = 125\\\\ \hookrightarrow\sf a + 6d = 23\\\dfrac{\qquad\qquad\qquad \qquad \qquad}{}\\\hookrightarrow\sf (40d - 6d) = (125 - 23)\\\\\hookrightarrow\sf 34d = 102\\\\\hookrightarrow\sf d = \cancel\dfrac{102}{34}\\\\\hookrightarrow\red{\sf d = 3}$

❍ $\underline{\textsf{Putting the Value of d in eq(1)}}$

$\longmapsto\sf 23 = a + 6d\\\\ \longmapsto\sf 23 = a +6(3)\\\\ \longmapsto\sf 23 = a + 18\\\\ \longmapsto\sf 23 - 18 = a\\\\ \longmapsto\green{\sf a = 5}$

$\bullet\:\underline\textsf{First Arithmetic Progresion :} \\\\ \dashrightarrow \sf a, (a + d), (a +2d), (a +3d)..\\\\\dashrightarrow\tt 5, (5 + 3), (5 + 2(3)),(5 + 3(3))..\\\\\dashrightarrow\sf\orange{ 5, 8,11,14..}$

$\bullet\:\underline\textsf{Second Arithmetic Progresion :} \\\\\dashrightarrow \sf a, (a + 2d), (a +2(2d)), (a +3(2d))..\\\\\dashrightarrow\tt 5, (5 + 2(3)), (5 + 2(6)), (5 + 3(6))..\\\\\dashrightarrow\sf\orange{ 5, 11,17, 23..}$

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