Progression level-3

If in any decreasing arithmetic progression, sum of all its terms, except for the first term, is equal to –36, the sum of all its terms, except for the last term, is zero, and the difference of the tenth and the sixth term is equal to –16, then what will be first term of this series?

The sum of all terms of the arithmetic progression having ten terms except for the first term, is 99, and except for the sixth term, 89. Find the third term of the progression if the sum of the first and the fifth term is equal to 10.

Product of the fourth term and the fifth term of an arithmetic progression is 456. Division of the ninth term by the fourth term of the progression gives quotient as 11 and the remainder as 10. Find the first term of the progression.

A number of saplings are lying at a place by the side of a straight road. These are to be planted in a straight line at a distance interval of 10 metres between two consecutive saplings. Mithilesh, the country’s greatest forester, can carry only one sapling at a time and has to move back to the original point to get the next sapling. In this manner he covers a total distance of 1.32 kms. How many saplings does he plant in the process if he ends at the starting point?

A geometric progression consists of 500 terms. Sum of the terms occupying the odd places is P₁ and the sum of the terms occupying the even places is P₂. Find the common ratio.

The sum of the first ten terms of the geometric progression is S₁ and the sum of the next ten terms (11th through 20th) is S₂. Find the common ratio.

 (a) (S₁/S₂)^1/10

(b) – (S₁/S₂)^1/10

(c) ±  (√(S₂/S₁))^1/10

(d) (S₁/S₂)^1/5

The first and the third terms of an arithmetic progression are equal, respectively, to the first and the third term of a geometric progression, and the second term of the arithmetic progression exceeds the second term of the geometric progression by 0.25. Calculate the sum of the first five terms of the arithmetic progression if its first term is equal to 2.

If (2 + 4 + 6 + … 50 terms)/(1 + 3 + 5 + … n terms) = 51/2, then find the value of n.

\( (\,\underbrace{666\ldots}_{n\ \text{digits}}\,)^2+(\,\underbrace{888\ldots}_{n\ \text{digits}}\,) \)

is equal to

(a) \( \dfrac{4}{9}(10^n-1) \)

(b) \( \dfrac{4}{9}(10^{2n}-1) \)

(c) \( \dfrac{4(10^n-10^{n-1}-1)}{9} \)

(d) \( \dfrac{4(10^n+1)}{9} \)

The interior angles of a polygon are in AP. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.