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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.

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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.

Let the value of certificates purchased in the first year be Rs. ‘a’ and annual increase = Rs.25. Here, first term = a, d = 25, S₂₀ = 7250.
We know that,

straight S subscript straight n equals straight n over 2 left square bracket 2 straight a space plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket rightwards double arrow space straight S subscript 10 equals 20 over 2 left square bracket 2 straight a plus left parenthesis 20 minus 1 right parenthesis 25 right square bracket rightwards double arrow space 7250 space equals space 10 space left parenthesis 2 straight a plus 19 cross times 25 right parenthesis

⇒ 7250 = 10 (2a + 475)
⇒ 725 = 2a + 475 ⇒ 2a = 250
⇒ a = 125
Hence, the value of certificates purchased in first year = Rs. 125.
Now, value of certificates purchased in 12th year.
a₁₂ = a + (12 – 1 )d = 125 + (11) (25) = 125 + 275 = Rs. 400.

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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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