Maths Q6 – JEE-MAIN 2026

Home Maths Sequences and SeriesQuestion #398

Let

A_{1}, A_{2}, A_{3}, . . ., A_{39}

be 39 arithmetic means between the numbers 59 and 159. Then the mean of

A_{25}, A_{28}, A_{31}

and

A_{36}

is equal to :

Clue 1 of 3: Conceptual

The arithmetic means inserted between two numbers, along with the two numbers themselves, form a single Arithmetic Progression (AP).

🥷

Ninja Strategy•Symmetry and Averaging

The mean of the given terms is the term at the average index, $A_{30}$ . This term lies halfway between the central term $A_{20}$ (value is the average of endpoints, 109) and the conceptual term $A_{40}$ (value is the endpoint, 159). Thus, the mean is $(109 + 159) / 2 = 134$ .

🧠 Full Solution Path

Step 1: Determine the Properties of the Arithmetic Progression✦ Active

The sequence $59, A_{1}, A_{2}, \dots, A_{39}, 159$ forms an Arithmetic Progression (AP). The first term is $a = 59$ and the last term is $b = 159$ . Since there are 39 arithmetic means, the total number of terms in the AP is $N = 39 + 2 = 41$ .

The 41st term is $T_{41} = 159$ . We can find the common difference $d$ using the formula $T_{N} = a + (N - 1) d$ .

159 = 59 + (41 - 1) d

100 = 40 d ⟹ d = \frac{100}{40} = 2.5

The formula for the $k$ -th arithmetic mean is $A_{k} = a + k d$ .

Step 2: Calculate the Mean of the Specified Terms○ Expand

We need to find the mean of $A_{25}, A_{28}, A_{31},$ and $A_{36}$ .

Mean = \frac{A_{25} + A_{28} + A_{31} + A_{36}}{4}

Substitute the formula $A_{k} = a + k d$ into the expression for the mean:

Mean = \frac{(a + 25 d) + (a + 28 d) + (a + 31 d) + (a + 36 d)}{4}

Simplify the expression:

Mean = \frac{4 a + (25 + 28 + 31 + 36) d}{4} = a + \frac{120}{4} d = a + 30 d

Now, substitute the values of $a = 59$ and $d = 2.5$ :

Mean = 59 + 30 (2.5) = 59 + 75 = 134

💡 Teacher's Secret Hint

Notice that the mean of a set of terms from an AP is equal to the term corresponding to the average of their indices. Here, the average index is $(25 + 28 + 31 + 36) / 4 = 30$ , so the mean is simply $A_{30}$ .

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Maths Question 6 – JEE-MAIN 2026

Clue 1 of 3: Conceptual

🧠 Full Solution Path