Tips on cracking Aptitude Questions on Progressions

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4 TIPS on cracking Aptitude Questions on
Progressions
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Tip 1 Sum of n terms of an AP n x
(Arithmetic Mean of first and last terms)

• If a be the first term of an AP and l be the last
  term, i.e., the nth term, then the sum of the AP
  will be n(a l)/2.
• If the last term is not specified, replace l by a
  (n 1)d where d is the common difference. Then
  the sum of the AP will be n 2a (n 1)d
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• Question 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.
• Solution
• Let there be n sides.
• Then the largest side is 120 (n 1)5 degrees
  115 5n
• Sum of interior angles n(120 115 5n)/2
  n(235 5n)/2
• In any polygon, the sum of the interior angles is
  (n 2) x 180
• n(235 5n)/2 (n 2) x 180
• n2 25n 1440
• (n 9)(n 16) 0
• n 9 or 16
• If n16, largest angle 115 5 x16 195
• But the interior angle of a polygon cannot be
  greater than 180 .
• Hence, n 9.

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Tip 2 If the terms of an AP are operated upon
by some constant, the resultant AP bears a
certain relation to the previous one.

• If each term of an AP is multiplied by C, common
  difference becomes dC and the sum SC
• If a constant C is added to each term of an AP
  the new sum is SnC and d remains same
• Question 6 kids are born into a family and their
  age difference is 3 years. If the current age of
  the youngest child is 4 years, what will be the
  sum of their ages 5 years from now?
• Solution
• According to the question, age of the eldest
  child 4 5 3 19 yrs
• Sum of their ages now 6(19 4)/2 69 years.
• Sum of their ages 5 years from now 69 (6 x 5)
  99 years.
• Question The sum of 10 numbers in an AP is 810.
  What is their sum when all the numbers are
  divided by 3?
• Solution
• New sum 810/3 270.

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Tip 3 Wherever possible, use the Formulae for
Sum

• Sum of first n natural numbers, 1, 2, 3, ., n
  is
• S n(n1)/2
• Sum of squares of first n natural numbers 12,
  22, 32, ., n2 is
• S n(n1)(2n1)/6
• Sum of cubes of first n natural numbers, 13,
  23, 33, ., n3 is
• S n2(n1)2/4
• If the nth term of a sequence is Tn an3 bn2
  cn d, then the sum of the series is
• S aSn3 bSn2 cSn nd
• Question Find the sum of n 2(n 1) 3(n
  2) . (n 1)2 n
• Solution
• The rth term of the series can be written as Tr
  r(n r 1)
• (n1)r r2.
• Thus, the sum of the series will be
• S (n1)Sr Sr2
• (n1)Sn Sn2
• (n1).n.(n1)/2 n(n1)(2n1)/6
• n(n1)(3n3 2n 1)/6
• n(n1)(n2)/6.

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Tip 4 When the quantities are in progression,
choose the variable as the middle element to make
your calculations easier.

• Question The sum of ages of 5 children born at
  the intervals of 3 years each is 50 years. What
  is the age of the youngest child?
• Solution
• Let the age of the 3rd child be a.
• Then the successive ages of the children are
  (a-6), (a-3), a, (a3), (a6).
• According to the question,
• 5a50 (Since the positive and
  negative terms are exactly equal and thus cancel
  out.)
• a10
• Age of youngest child a-6 4 years
• Question A, B and C are neighbors. If As age is
  thrice that of B and Bs age is thrice that of C
  and the product of their ages is 216, what is Cs
  age?
• Solution
• Let Bs age be b. Then,
• 3b b (b/3) 216
• b3 216 (Since the multipliers and divisors
  cancel each other)
• b (216)1/3 6
• Cs age b/3 2 years.

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