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Progression Examples

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Examples :

(1) Find 10^th term of the A.P. 351, 357, 363, 369, …

Solution. For given A.P., we have : a = 351 and d = 6.

Here, $T_{n}$ = 6n + 345

$\therefore T_{10}=6(10)+345=405$

(2) How many terms are there in the A.P. 201, 208, 215, …, 362 ?

Solution. For given A.P., $T_{n}$ = 7n + 244

$\therefore 7n+244=362 \Rightarrow n=24$

(3) How many three digit numbers are there which are divisible by 13 ?

Solution. The requisite numbers are : 104, 117, 130, 143, …, 988.

For given A.P., $T_{n}$ = 13n + 91

$\therefore 13n+91=988 \Rightarrow n=69$

(4) Find the sum of all even numbers lying between 9 and 99.

Solution. The requisite numbers are : 10, 12, 14, 16, . . . , 98

Here, $T_{n}=2n+8$

$\therefore 2n+8=98$

$\therefore n=45$

$\therefore$ Required sum $= \frac{n}{2}\left (a+l \right )=\frac{45}{2}\times \left (10+98 \right )=2430$

(5) Find three arithmetic means between 7 and 47.

Solution. Now, $d= \frac{b-a}{n+1}$ : Where, a = 7, b = 47 & n = 3

$\therefore d= \frac{47-7}{3+1}$

$= \frac{40}{4}$

= 10

$\therefore$ Nos. are 7 + 10 = 17, 17 + 10 = 27, 27 + 10 = 37, i.e. Three arithmetic means are 17, 27 and 37.