Logo
Jeeify
MCQEasyJEE 2026Arithmetic Progression (AP)

JEE Mathematics 2026 Question with Solution

The common difference of the A.P.: a1,a2,…,ama_1, a_2, \ldots, a_m is 1313 more than the common difference of the A.P.: b1,b2,…,bnb_1, b_2, \ldots, b_n. If b31=−277b_{31} = -277, b43=−385b_{43} = -385 and a78=327a_{78} = 327, then a1a_1 is equal to:

  • A

    1616

  • B

    1919

  • C

    2424

  • D

    2121

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: Two arithmetic progressions with common differences dad_a and dbd_b such that da=db+13d_a = d_b + 13. Also, b31=−277b_{31} = -277, b43=−385b_{43} = -385, and a78=327a_{78} = 327.

Find: a1a_1.

For an arithmetic progression,

an=a1+(n−1)da_n = a_1 + (n-1)d

Using the A.P. bnb_n:

b31=b1+30db=−277b_{31} = b_1 + 30d_b = -277 b43=b1+42db=−385b_{43} = b_1 + 42d_b = -385

Subtracting the first equation from the second,

12db=−10812d_b = -108 db=−9d_b = -9

Now use the relation between the common differences:

da=db+13=−9+13=4d_a = d_b + 13 = -9 + 13 = 4

For the A.P. ana_n,

a78=a1+77daa_{78} = a_1 + 77d_a 327=a1+77(4)327 = a_1 + 77(4) 327=a1+308327 = a_1 + 308 a1=19a_1 = 19

Therefore, the correct option is B and a1=19a_1 = 19.

Using term difference first

Given: b31=−277b_{31} = -277 and b43=−385b_{43} = -385 in one A.P., and the common difference of the other A.P. is 1313 more.

Find: a1a_1.

In an arithmetic progression, the difference between the 43rd43^{\text{rd}} term and the 31st31^{\text{st}} term is

b43−b31=(43−31)dbb_{43} - b_{31} = (43-31)d_b

So,

−385−(−277)=12db-385 - (-277) = 12d_b −108=12db-108 = 12d_b db=−9d_b = -9

Hence,

da=−9+13=4d_a = -9 + 13 = 4

Now use a78=a1+77daa_{78} = a_1 + 77d_a:

327=a1+77×4327 = a_1 + 77 \times 4 327=a1+308327 = a_1 + 308 a1=19a_1 = 19

Therefore, the first term is 1919, so the correct option is B.

Common mistakes

  • A common mistake is taking the difference relation in the wrong order and writing da=db−13d_a = d_b - 13. This is wrong because the question says the common difference of the A.P. a1,a2,…,ama_1, a_2, \ldots, a_m is 1313 more than that of b1,b2,…,bnb_1, b_2, \ldots, b_n. Use da=db+13d_a = d_b + 13 instead.

  • Students often use the wrong term formula, such as an=a1+nda_n = a_1 + nd. This is wrong for an arithmetic progression because the correct formula is an=a1+(n−1)da_n = a_1 + (n-1)d. Therefore, for a78a_{78}, the multiplier of the common difference must be 7777, not 7878.

  • Another mistake is subtracting the equations for b31b_{31} and b43b_{43} incorrectly. The term index gap is 43−31=1243 - 31 = 12, so the common-difference equation must be 12db=−10812d_b = -108. Do not treat it as 13db13d_b or forget the negative sign in −385−(−277)-385 - (-277).

Practice more Arithmetic Progression (AP) questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Arithmetic Progression (AP)

Related questions