In an arithmetic progression, the sum of the first four terms is $54$ and the sum of the first seven terms is $147$ . What is the value of the tenth term?

hard MCQ 3 marks Class 10 · Mathematics · Arithmetic Progressions

Answer

A. 51

The correct answer is A. 51 — Hence the correct option is A. 51.

Step-by-step solution

Write the sum formula for $n$ terms: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .
For $n = 4$ : $S_{4} = 2 [2 a + 3 d] = 54 \Rightarrow 2 a + 3 d = 27$ … (1).
For $n = 7$ : $S_{7} = \frac{7}{2} [2 a + 6 d] = 147 \Rightarrow 2 a + 6 d = \frac{147 \times 2}{7} = 42$ … (2).
Subtract (1) from (2): $(2 a + 6 d) - (2 a + 3 d) = 3 d = 42 - 27 = 15$ giving $d = 5$ .
Substitute $d$ into (1): $2 a + 3 \times 5 = 27 \Rightarrow 2 a + 15 = 27 \Rightarrow 2 a = 12 \Rightarrow a = 6$ .
The $1 0^{th}$ term is $a_{10} = a + 9 d = 6 + 9 \times 5 = 6 + 45 = 51$ .
Hence the correct option is A. 51.

Common mistake: Mixing up the coefficients while writing

S_{n}

; for

S_{7}

many forget to multiply by

\frac{7}{2}

correctly.

Concept tested: finding specific term using given sums