NCERT Class 10 Arithmetic Progressions problem: each of the 3 sections of Classes I to XII plants trees equal to its class number. Find the total number of trees planted.
NCERT Class 10 Arithmetic Progressions problem: a spiral of 13 successive semicircles with radii increasing by 0.5cm each time. Find its total length using the AP sum formula.
Given the sum of the first 20 terms of an AP is 30 and the sum of the first 30 terms is 20, find S50 by solving for the first term and common difference.
m arithmetic means are inserted between 1 and 31 so the ratio of the 7th and (m−1)th means is 5:9. Find m by expressing both means via the common difference D=30/(m+1).
For an infinite GP a, ar, ar², … with Σar^n=57 and Σa³r^3n=9747, find a+18r by dividing the cubed-series sum by the cube of the original sum.
Find the greatest positive integer k such that 49^k+1 divides 1+49+49²+…+49¹²⁵, by writing the sum as a GP and factoring it as a difference of squares.
In an arithmetic progression, if m times the mth term equals n times the nth term (m≠n), prove that the (m+n)th term must equal zero.
Find the positive integer n satisfying 2·2²+3·2³+…+n·2ⁿ=2^(n+10), by summing the series via the standard shift-and-subtract technique for a sum of the form Σk·rᵏ.
Given positive reals a,b,c in AP with abc=4, find the minimum value of b using the AM-GM inequality on a, b, c.
Find the sum of the alternating series 1/(1·2)−1/(2·3)+1/(3·4)−… to infinity, by splitting each term via partial fractions and matching against the log(1+x) expansion.