Solved Math Problems - Sum of first 272 even numbers, Coefficient of x6, Geometric sequence, Club member selection

This document provides solutions to four Math problems related to Sum of first 272 even numbers, Coefficient of x6, Geometric sequence, Club member selection. It includes step-by-step explanations and formulas used to arrive at the answers.

4)
Sum of first 272 natural even numbers.
Answer:B) 74,256
The sum can be written as:
S = 2 + 4 + 6 + 8 + · · · + 270 + 272 + . . . 542 + 544
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[Take the common factor 2 out]
= 2{1 + 2 + 3 + 4 + · · · + 135 + 136 + · · · + 271 + 272}
[Sum inside the bracket is the sum of first 272 natural numbers]
[Sum of n consecutive natural numbers is :
n(n + 1)
2 ]
S = 2 ×n(n + 1)
2
= n(n + 1)
[for n = 272]
S = 272 × 273 = 74256
5)
Coefficient of x6
Answer:B) 17,010
The Binomial expansion of (ax + b)n is:
(ax + b)n =
nX
k=0
n
k an−kxn−kbk
where, n
k an−kbk is the coefficient ofxn−k. The binomialexpansion of
(x + 3)10 is:
(x + 3)10 =
10X
k=0
10
k x10−k3k
for k = 4, the power of x is 6.Thus, when k = 4, the coefficient of x6 is:
C4 = 10
4 34
10
4 = 10!
4! × 6!
= 10 × 9 × 8 × 7
4 × 3 × 2 = 210
34 = 81
Therefore,
C4 = 210 × 81 = 17010
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6)
Answer:C) 3279
The geometric sequence is:
3, 9, 27, . . .
The first term in the sequence is:a1 = 3.
Common ratio (r) is the ration of consecutive terms in the sequence:
r = 9
3 = 27
9 = · · · = 3
Sum of the first n terms in a geometric sequence is:
Sn = a1 + a2 + · · · + an
It is obtained using the formula:
Sn = a1 · rn − 1
r − 1
For the given sequence, the sum of first 7 terms is:
S7 = 3 ×37 − 1
3 − 1= 3279
7)
Answer:A) 720
From a club of 10 members any one can be president.So, the president can
be selected in 10 different ways.
Since the president cannot be selected again as vice-president, only 9 mem-
bers remain to be selected from.From the 9 members, a vice-president can
be selected in 9 different ways.
Similarly,out the remaining 8 members,a secretary can be selected in 8
different ways.
Therefore, out a club of 10 members, president, vice-president and secretary
can be selected in: 10P3 = 10 × 9 × 8 = 720
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