If the coefficient of the three successive terms in the binomial expansion of $$(1 + x)^n$$ are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is

We consider the binomial expansion of $$\bigl(1+x\bigr)^n.$$

The general term (counting from 0) is written first:

$$T_{r+1}= \binom{n}{r}\,x^{\,r},$$

where $$r=0,1,2,\dots,n.$$ The numerical coefficient of this term is simply $$\binom{n}{r}.$$

According to the question, the numerical coefficients of three successive terms are in the ratio $$1:7:42.$$ Let these three successive terms correspond to the indices $$r,\;r+1,\;r+2.$$ Then we have

$$\binom{n}{r} : \binom{n}{r+1} : \binom{n}{r+2}=1:7:42.$$

To translate the verbal ratio into equations, we equate the successive ratios of these coefficients:

First ratio:

$$\frac{\binom{n}{r+1}}{\binom{n}{r}}=7.$$

Second ratio:

$$\frac{\binom{n}{r+2}}{\binom{n}{r+1}}=\frac{42}{7}=6.$$

Now we recall the standard identity for consecutive binomial coefficients:

$$\frac{\binom{n}{k+1}}{\binom{n}{k}}=\frac{n-k}{k+1}.$$

Applying this identity to each of the above ratios, we obtain two algebraic equations.

For the first ratio:

$$\frac{\binom{n}{r+1}}{\binom{n}{r}}=\frac{n-r}{r+1}=7.$$

Hence

$$n-r=7(r+1).$$

Simplifying,

$$n-r=7r+7 \;\;\Longrightarrow\;\; n=8r+7.$$

For the second ratio we write

$$\frac{\binom{n}{r+2}}{\binom{n}{r+1}}=\frac{n-(r+1)}{r+2}=6.$$

Simplifying the numerator gives

$$\frac{n-r-1}{r+2}=6.$$

Cross-multiplying, we find

$$n-r-1=6(r+2).$$

Expanding the right side,

$$n-r-1=6r+12.$$

Hence

$$n=7r+13.$$

We now have two expressions for $$n:$$

$$n=8r+7 \quad\text{and}\quad n=7r+13.$$

Equating these,

$$8r+7=7r+13.$$

Subtracting $$7r$$ from both sides,

$$r+7=13.$$

So

$$r=6.$$

Substituting $$r=6$$ back into either expression for $$n,$$ say $$n=8r+7,$$ we get

$$n=8(6)+7=48+7=55.$$

Thus the three successive terms are the ones with indices

$$r=6,\; r+1=7,\; r+2=8.$$

The first of these, $$r=6,$$ corresponds to the term

$$T_{r+1}=T_{6+1}=T_7.$$

Since term counting in binomial expansions starts with $$T_1$$ (the $$r=0$$ term), $$T_7$$ is called the 7th term of the expansion.

Hence, the correct answer is Option D.