= Solution
Interpret the connectivity hypothesis in its usual based sense: $X$ is $(n-2)$-connected, with $n\ge2$ for the nontrivial suspension range. Form the suspension as two cones $C_-X$ and $C_+X$ meeting in $X$. Both cones are <contractible>. Their relative <homotopy> sequences give $\pi_j(C_\pm X,X)\cong\pi_{j-1}(X)$ in the relevant degrees, so both cone pairs are $(n-1)$-connected.
The <homotopy excision theorem> says that if two <CW pairs> meeting in a connected subcomplex are $p$- and $q$-connected, $p,q\ge1$, the map from the first relative pair to the union relative to the second part is an <isomorphism> in degrees $i<p+q$ and a surjection in degree $p+q$. Here $p=q=n-1$, giving
$$
\pi_i(C_+X,X)\longrightarrow\pi_i(\Sigma X,C_-X)
$$
as an <isomorphism> for $i<2n-2$ and a surjection for $i=2n-2$. Contractibility identifies the domain with $\pi_{i-1}(X)$ and the target with $\pi_i(\Sigma X)$. The comparison under these identifications is the suspension map: a <sphere> representative is coned in each half and the two cones give its suspended <sphere> representative. Hence the <Freudenthal suspension from two cones> proves
$$
\boxed{\pi_{i-1}(X)\to\pi_i(\Sigma X)\text{ is onto for }i\le2n-2,\text{ and is an isomorphism for }i<2n-2.}
$$
For connected <CW complexes>, reduced and unreduced suspension have the same based <homotopy> type, so either standard suspension convention gives this result. In the low-dimensional range, <pointed sets> are used where necessary; the displayed positive-degree <homomorphism> assertion concerns the usual <groups>.
For $S^r$ with $r\ge2$, cellular approximation makes every <sphere> map of degree below $r$ <null-homotopic>: it can be homotoped into the point skeleton of the CW structure having cells only in degrees zero and $r$. The <Hurewicz theorem> therefore identifies its first possible nonzero <homotopy group> with <homology>. Its <cellular chain complex> gives $H_r(S^r;\mathbb Z)=\mathbb Z$. For $r=1$, lifting a based loop to the <universal cover> $\mathbb R\to S^1$ classifies it by an integer endpoint displacement, so $\pi_1(S^1)=\mathbb Z$; the degree-one Hurewicz map is <abelianization> and is an <isomorphism> here. Thus
$$
\boxed{\pi_r(S^r)\cong H_r(S^r;\mathbb Z)\cong\mathbb Z\qquad(r\ge1).}
$$
Alternatively, after the $r=2$ Hurewicz base case, the suspension <isomorphisms> above propagate the generator to every higher <sphere>.
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