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Let ‘a’ be the 1st term and ‘d’ be the common difference of the A .P. ma m = na n ⇒ m[a + (m – 1) d] = n[a + (n – 1) d] ⇒ m[a + md – d) = n[a + nd – d] ⇒ ma + m 2 d – md = na + n 2 d – nd ⇒ ma – na + m 2 d – n 2 d – md + nd = 0 ⇒ (m – n)a + (m 2 – n 2 )d – d(m – n) =0 ⇒ (m – n)a + (m + n) (m – n)d – (m – n)d = 0 ⇒ (m – n) [a + (m + n)d – d] = 0 ⇒ a + (m + n)d – d = 0 ⇒ a + (m + n – 1)d = 0 ⇒ a m + n = 0 Thus, the (m + n)th term of the given A .P. is zero.

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Arithmetic Progressions

Short Answer Type

291. A sum of Rs. 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these, interests form an A .P. ? If so, find the interest at the end of 30 yrs.

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292. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third and so on. There are 5 rose plant in the last row. How many rows are there in the flower bed?

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293. If m times the mth term of an A .P. is equal to n times its nth term show that (m + n)th term of the A .P. is zero.

Let ‘a’ be the 1st term and ‘d’ be the common difference of the A .P.
ma_m = na_n⇒ m[a + (m – 1) d] = n[a + (n – 1) d]
⇒ m[a + md – d) = n[a + nd – d]
⇒ ma + m²d – md = na + n²d – nd
⇒ ma – na + m²d – n²d – md + nd = 0
⇒ (m – n)a + (m² – n²)d – d(m – n) =0
⇒ (m – n)a + (m + n) (m – n)d – (m – n)d = 0
⇒ (m – n) [a + (m + n)d – d] = 0
⇒ a + (m + n)d – d = 0 ⇒ a + (m + n – 1)d = 0
⇒ a_{m + n} = 0
Thus, the (m + n)th term of the given A .P. is zero.

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Long Answer Type

294. If the mth term of an A .P. is 1/n and the nth term is 1/m, find the (mn)th term.

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Short Answer Type

295. Split 69 into three parts such that they are in A .P. and the product of two smaller parts is 483.

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296. The ratio of the 7th to the 3rd term of an A .P. is 12 : 5. Find the ratio of the 13th to 4th term.

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297. If the Pth term of an A .P. is q and qth term is P, prove that its nth term is (P + q – n).

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298. Find the 10th term from the end of the A .P. 8, 10, 12................ 126

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299. The sum of the 4th and 8th terms of an A .P. is 24 and the sum of the sixth and tenth terms is 44. Find the First three terms of the A .P.

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300.

The first and the last term of an A .P. are 4 and 81 respectively. If the common difference is 7, how many terms are there in the A .P. and what is their sum?

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