If the 5th term of an Arithmetic Progression is 15 and the 9th term is 27, find the common differenc

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If the 5th term of an Arithmetic Progression is 15 and the 9th term is 27, find the common difference.

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{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1:","type":"text"},{"text":" Let the first term of the AP be ","type":"text"},{"attrs":{"latex":"a"},"type":"math_inline"},{"text":" and the common difference be ","type":"text"},{"attrs":{"latex":"d"},"type":"math_inline"},{"text":". The ","type":"text"},{"attrs":{"latex":"n"},"type":"math_inline"},{"text":"-th term is ","type":"text"},{"type":"math_inline","attrs":{"latex":"a_n = a + (n-1)d"}},{"text":".","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2:","type":"text"},{"text":" Write equations for the given terms:","type":"text"}],"type":"paragraph"},{"content":[{"type":"list_item","content":[{"content":[{"attrs":{"latex":"a_5 = a + 4d = 15"},"type":"math_inline"}],"type":"paragraph"}]},{"content":[{"content":[{"attrs":{"latex":"a_9 = a + 8d = 27"},"type":"math_inline"}],"type":"paragraph"}],"type":"list_item"}],"type":"ordered_list","attrs":{"start":1}},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3:","type":"text"},{"text":" Subtract the first equation from the second equation:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"(a + 8d) - (a + 4d) = 27 - 15 \implies 4d = 12"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4:","type":"text"},{"type":"text","text":" Solve for "},{"attrs":{"latex":"d"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"d = \frac{12}{4} = 3"},"type":"math_block"},{"content":[{"text":"So the common difference is ","type":"text"},{"attrs":{"latex":"3"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"}],"type":"doc"}

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