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Arithmetic Progressions

Long Answer Type

311. The ratio of the sums of and ‘n’ terms of an A .P. is m² : n². Show that the ratio of the mth and nth terms is (2m – 1) : (2n – 1).

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Short Answer Type

312. The interior angles of a polygon are in A .P. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon. Sol. Let ‘n’ be the sides of the polygon. We know that some of all interior angies of a polygon = (n - 2) x

180 degree

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313. Find the sum of the series :

4 + 8 + 12 +......... + 100.

Let ‘b’ be the first term and ‘d’ the common difference of the A .P.
a = 4, d = 4, a_n= 100 = 1
We know that, a_n= a+ (n -1)d

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100 = 4 + (n - 1) d

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100 = 4 + (n - 1) 4

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4n = 100

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n = 25

And comma space space straight S subscript straight n equals straight n over 2 left square bracket straight a plus straight l right square bracket space space space left square bracket because space straight l space equals straight T subscript straight n right parenthesis right square bracket So comma space space space space straight S subscript 25 equals 25 over 2 space space space left square bracket 4 space plus space 100 right square bracket space space space space space space space space space space space space space space equals 25 over 2 cross times 104 equals 25 cross times 52 space space space space space space space space space space space space space space space equals space 1300

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314. Find the sum of the series :

7 + 10 1/2 + 14 + .. .. + 84.

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315. Find the sum of the series :
34 + 32 + 30 +........ + 10

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316. Find the sum of the series :
–5 + (–8) + (–11) +........ + (–230).

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317. Find the sum of all natural numbers between 100 and 1000 which arc multiples of 7.

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318.

Tanvika purchases every year National Saving Certificates of value exceeding the last year purchases by Rs. 25. After 20 years, he find the total value of certificates purchased by him is Rs. 7250. Find the value of the certificates purchased.

(i) in the first year. (ii) in the 12th year.

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319. Two motorist A and B starts together in the same direction from New Delhi to Mumbai. A travels with the uniform speed of 100 km/hr and B travels 50 km/hr in the first hour and increases the speed by 5 km/hr. After how many hour will ‘B’ overtake A, if both motorist never stops.

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320. Nimit saves Rs. 50 during a month, Rs. 100 during next month, Rs. 150 in next month, if he continues his savings in this way, in how many months will he save Rs. 10500.

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