1If \( lr - a = 90 \) and \( r - 1 = 9 \), find \( s_n \).
2Write the sum of the first 5 terms of the geometric series if a = 1 and r = 2.
3If \( 500 = \frac{1893 \times 3 - a}{3 - 1} \), find the value of a.
4In the geometric series 2 + 4 + 8 + 16 + 32 + 64 + 128, what is the common ratio?
5Write the first term and the last term of the GP 81 + 27 + 9 + 3 + 1 + \( \frac{1}{3} \).
6Find the sum of the geometric series 24 + 12 + 6 + ... to 10 terms.
7Find the sum of the geometric series 3 - 6 + 12 - ... to 7 terms.
8Find the sum of the geometric series 27 + 18 + 12 + ... to 6 terms.
9Find the sum of the geometric series \( \sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + \cdots \) to 8 terms.
10Find the sum of the geometric series 2 + 6 + 18 + ... + 4374.
11Find the sum of the geometric series 3 + 6 + 12 + ... + 768.
12The first term of a GP is 1 and the sum of the third and fifth terms is 90. Find the common ratio.
13The sum of the first two terms of a GP is 10 and the first term is 2. Find the common ratio.
14The sum of the first 4 terms of a GP whose common ratio is 2 is 255. Find the first term.
15The sum of a series in GP whose common ratio is 4 is 1364, and the last term is 1024. Find the first term.
16The first term and common ratio of a geometric series are \( \frac{1}{3} \) and 3. Find the sum of the first 6 terms.
17How many terms are there in a GP where a = 2, r = 3 and the sum is 728?
18How many terms of the series 5 + 10 + 20 + ... must be taken so that the sum is 315?
19Find the sum of the first 8 terms of a GP whose 3rd and 7th terms are 8 and 128.
20In a GP, the 4th and 7th terms are 27 and 729. Find (i) the common ratio and first term, (ii) the geometric series, (iii) the sum of the first eight terms.
21In a GP, the sum of the first two terms is 18 and the sum of the first four terms is 90. Find (i) the common ratio and first term, (ii) the geometric series, (iii) the sum of the first 6 terms.
22In a GP, the sum of the first three terms is \( \frac{7}{4} \) and the sum of the first six terms is \( \frac{63}{4} \). Find the sum of the first 8 terms.
23In a GP, the sum of three numbers is 28 and their product is 512. Find the numbers.
24The sum of three consecutive terms of a GP is 13 and their product is 27. Find the three terms.
25The sum of three numbers in GP is 56. If 1, 7 and 21 are subtracted from them in order, the results form consecutive terms of an AP. Find the original numbers.
26The sum of three numbers in AP is 15. If 1 and 5 are added to the second and third numbers, the first number together with these two new numbers are in GP. Find the original numbers.
27The IQ scores of 10 students follow a rule: the second student's score is double the first's, the third's is double the second's, and so on. If the first student's score is 2, find the score of the 10th student and the total score of all 10 students.
28Why can the formula \( s_n = \frac{a(r^n - 1)}{r - 1} \) not be used when r = 1, and what is the sum in that case?
29Show that the two forms \( \frac{a(1 - r^n)}{1 - r} \) and \( \frac{a(r^n - 1)}{r - 1} \) are equal.
30In the derivation of the sum formula, why do almost all the terms cancel when \( rs_n \) is subtracted from \( s_n \)?
31Four numbers in GP are taken as \( \frac{a}{r^3}, \frac{a}{r}, ar, ar^3 \). What is their common ratio, and what is their product?