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CAT 2022 Slot 3 — QA Question 13

Arithmetic ProgressionEasy

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

Answer & solution

Correct answer: 548

Answer: 548

Solution

Easy

This is an AP with first term 3838 and common difference 1717. The average of any AP equals the average of its first and last terms — so we only need the smallest and largest 33-digit members.

1

Set up the general term.

Tn=38+(n−1)×17T_n=38+(n-1)\times 17
2

Smallest 3-digit term (need Tn≥100T_n\ge 100):

38+(n−1)×17≥100⇒ n−1≥6217≈3.65⇒n≥5⇒ T5=38+4×17=106\begin{aligned} &38+(n-1)\times 17\ge 100\\ &\Rightarrow\ n-1\ge\frac{62}{17}\approx 3.65\Rightarrow n\ge 5\\ &\Rightarrow\ T_5=38+4\times 17=106 \end{aligned}
3

Largest 3-digit term (need Tn≤999T_n\le 999):

38+(n−1)×17≤999⇒ n−1≤96117≈56.5⇒n≤57⇒ T57=38+56×17=990\begin{aligned} &38+(n-1)\times 17\le 999\\ &\Rightarrow\ n-1\le\frac{961}{17}\approx 56.5\Rightarrow n\le 57\\ &\Rightarrow\ T_{57}=38+56\times 17=990 \end{aligned}
4

Average = mean of the extremes:

106+9902=10962=548\frac{106+990}{2}=\frac{1096}{2}=548

The average is 548\mathbf{548}.

For any AP, average =first+last2=\dfrac{\text{first}+\text{last}}{2} — you never need to count how many terms there are or add them all up.

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CAT 2022 Slot 3 QA Q13: The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is — Solution | TheCATExam