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Examples 1 to 4 of progression

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

Soln. 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 ?

Soln. For given A.P., $T_n$ = 7n + 244

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

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

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

For given A.P., $T_n$ = 13n + 91

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

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

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

Here, $T_n$ = 2n + 8

$\therefore$ 2n + 8 = 98

$\therefore$ n = 45

$\therefore$ Required sum = $\frac{n}{2} ( a + l )$

= $\frac{45}{2} \times ( 10 + 98 )$

= 2430.